topology.sheaves.sheaf_condition.sites ⟷ Mathlib.Topology.Sheaves.SheafCondition.Sites

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -158,7 +158,8 @@ variable {X : TopCat} {ι : Type _}
 
 #print TopCat.Opens.coverDense_iff_isBasis /-
 theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
-    CoverDense (Opens.grothendieckTopology X) B ↔ Opens.IsBasis (Set.range B.obj) :=
+    CategoryTheory.Functor.IsCoverDense (Opens.grothendieckTopology X) B ↔
+      Opens.IsBasis (Set.range B.obj) :=
   by
   rw [opens.is_basis_iff_nbhd]
   constructor; intro hd U x hx; rcases hd.1 U x hx with ⟨V, f, ⟨i, f₁, f₂, hc⟩, hV⟩
@@ -170,7 +171,7 @@ theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
 
 #print TopCat.Opens.coverDense_inducedFunctor /-
 theorem coverDense_inducedFunctor {B : ι → Opens X} (h : Opens.IsBasis (Set.range B)) :
-    CoverDense (Opens.grothendieckTopology X) (inducedFunctor B) :=
+    CategoryTheory.Functor.IsCoverDense (Opens.grothendieckTopology X) (inducedFunctor B) :=
   (coverDense_iff_isBasis _).2 h
 #align Top.opens.cover_dense_induced_functor TopCat.Opens.coverDense_inducedFunctor
 -/
@@ -249,7 +250,8 @@ def isTerminalOfEqEmpty (F : X.Sheaf C) {U : Opens X} (h : U = ⊥) :
     is equivalent to a homomorphism between their restrictions to the indexing type
     `ι` of `B`, with the induced category structure on `ι`. -/
 def restrictHomEquivHom : ((inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.1) ≃ (F ⟶ F'.1) :=
-  @CoverDense.restrictHomEquivHom _ _ _ _ _ _ _ _ (Opens.coverDense_inducedFunctor h) _ F F'
+  @CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom _ _ _ _ _ _ _ _
+    (Opens.coverDense_inducedFunctor h) _ F F'
 #align Top.sheaf.restrict_hom_equiv_hom TopCat.Sheaf.restrictHomEquivHom
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,9 +3,9 @@ Copyright (c) 2021 Justus Springer. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Justus Springer
 -/
-import Mathbin.CategoryTheory.Sites.Spaces
-import Mathbin.Topology.Sheaves.Sheaf
-import Mathbin.CategoryTheory.Sites.DenseSubsite
+import CategoryTheory.Sites.Spaces
+import Topology.Sheaves.Sheaf
+import CategoryTheory.Sites.DenseSubsite
 
 #align_import topology.sheaves.sheaf_condition.sites from "leanprover-community/mathlib"@"4280f5f32e16755ec7985ce11e189b6cd6ff6735"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,16 +2,13 @@
 Copyright (c) 2021 Justus Springer. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Justus Springer
-
-! This file was ported from Lean 3 source module topology.sheaves.sheaf_condition.sites
-! leanprover-community/mathlib commit 4280f5f32e16755ec7985ce11e189b6cd6ff6735
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.CategoryTheory.Sites.Spaces
 import Mathbin.Topology.Sheaves.Sheaf
 import Mathbin.CategoryTheory.Sites.DenseSubsite
 
+#align_import topology.sheaves.sheaf_condition.sites from "leanprover-community/mathlib"@"4280f5f32e16755ec7985ce11e189b6cd6ff6735"
+
 /-!
 
 # Coverings and sieves; from sheaves on sites and sheaves on spaces

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -105,7 +105,7 @@ def presieveOfCovering {ι : Type v} (U : ι → Opens X) : Presieve (iSup U) :=
     then we get back the original presieve `R`. -/
 @[simp]
 theorem covering_presieve_eq_self {Y : Opens X} (R : Presieve Y) :
-    presieveOfCoveringAux (coveringOfPresieve Y R) Y = R := by ext (Z f);
+    presieveOfCoveringAux (coveringOfPresieve Y R) Y = R := by ext Z f;
   exact ⟨fun ⟨⟨_, _, h⟩, rfl⟩ => by convert h, fun h => ⟨⟨Z, f, h⟩, rfl⟩⟩
 #align Top.presheaf.covering_presieve_eq_self TopCat.Presheaf.covering_presieve_eq_self
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -40,23 +40,28 @@ namespace TopCat.Presheaf
 
 variable {X : TopCat.{w}}
 
+#print TopCat.Presheaf.coveringOfPresieve /-
 /-- Given a presieve `R` on `U`, we obtain a covering family of open sets in `X`, by taking as index
 type the type of dependent pairs `(V, f)`, where `f : V ⟶ U` is in `R`.
 -/
 def coveringOfPresieve (U : Opens X) (R : Presieve U) : (Σ V, { f : V ⟶ U // R f }) → Opens X :=
   fun f => f.1
 #align Top.presheaf.covering_of_presieve TopCat.Presheaf.coveringOfPresieve
+-/
 
+#print TopCat.Presheaf.coveringOfPresieve_apply /-
 @[simp]
 theorem coveringOfPresieve_apply (U : Opens X) (R : Presieve U) (f : Σ V, { f : V ⟶ U // R f }) :
     coveringOfPresieve U R f = f.1 :=
   rfl
 #align Top.presheaf.covering_of_presieve_apply TopCat.Presheaf.coveringOfPresieve_apply
+-/
 
 namespace CoveringOfPresieve
 
 variable (U : Opens X) (R : Presieve U)
 
+#print TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieck /-
 /--
 If `R` is a presieve in the grothendieck topology on `opens X`, the covering family associated to
 `R` really is _covering_, i.e. the union of all open sets equals `U`.
@@ -72,9 +77,11 @@ theorem iSup_eq_of_mem_grothendieck (hR : Sieve.generate R ∈ Opens.grothendiec
   obtain ⟨V, iVU, ⟨W, iVW, iWU, hiWU, -⟩, hxV⟩ := hR x hxU
   exact ⟨⟨W, ⟨iWU, hiWU⟩⟩, iVW.le hxV⟩
 #align Top.presheaf.covering_of_presieve.supr_eq_of_mem_grothendieck TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieck
+-/
 
 end CoveringOfPresieve
 
+#print TopCat.Presheaf.presieveOfCoveringAux /-
 /-- Given a family of opens `U : ι → opens X` and any open `Y : opens X`, we obtain a presieve
 on `Y` by declaring that a morphism `f : V ⟶ Y` is a member of the presieve if and only if
 there exists an index `i : ι` such that `V = U i`.
@@ -82,12 +89,16 @@ there exists an index `i : ι` such that `V = U i`.
 def presieveOfCoveringAux {ι : Type v} (U : ι → Opens X) (Y : Opens X) : Presieve Y := fun V f =>
   ∃ i, V = U i
 #align Top.presheaf.presieve_of_covering_aux TopCat.Presheaf.presieveOfCoveringAux
+-/
 
+#print TopCat.Presheaf.presieveOfCovering /-
 /-- Take `Y` to be `supr U` and obtain a presieve over `supr U`. -/
 def presieveOfCovering {ι : Type v} (U : ι → Opens X) : Presieve (iSup U) :=
   presieveOfCoveringAux U (iSup U)
 #align Top.presheaf.presieve_of_covering TopCat.Presheaf.presieveOfCovering
+-/
 
+#print TopCat.Presheaf.covering_presieve_eq_self /-
 /-- Given a presieve `R` on `Y`, if we take its associated family of opens via
     `covering_of_presieve` (which may not cover `Y` if `R` is not covering), and take
     the presieve on `Y` associated to the family of opens via `presieve_of_covering_aux`,
@@ -97,11 +108,13 @@ theorem covering_presieve_eq_self {Y : Opens X} (R : Presieve Y) :
     presieveOfCoveringAux (coveringOfPresieve Y R) Y = R := by ext (Z f);
   exact ⟨fun ⟨⟨_, _, h⟩, rfl⟩ => by convert h, fun h => ⟨⟨Z, f, h⟩, rfl⟩⟩
 #align Top.presheaf.covering_presieve_eq_self TopCat.Presheaf.covering_presieve_eq_self
+-/
 
 namespace PresieveOfCovering
 
 variable {ι : Type v} (U : ι → Opens X)
 
+#print TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology /-
 /-- The sieve generated by `presieve_of_covering U` is a member of the grothendieck topology.
 -/
 theorem mem_grothendieckTopology :
@@ -111,25 +124,32 @@ theorem mem_grothendieckTopology :
   obtain ⟨i, hxi⟩ := opens.mem_supr.mp hx
   exact ⟨U i, opens.le_supr U i, ⟨U i, 𝟙 _, opens.le_supr U i, ⟨i, rfl⟩, category.id_comp _⟩, hxi⟩
 #align Top.presheaf.presieve_of_covering.mem_grothendieck_topology TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology
+-/
 
+#print TopCat.Presheaf.presieveOfCovering.homOfIndex /-
 /-- An index `i : ι` can be turned into a dependent pair `(V, f)`, where `V` is an open set and
 `f : V ⟶ supr U` is a member of `presieve_of_covering U f`.
 -/
 def homOfIndex (i : ι) : Σ V, { f : V ⟶ iSup U // presieveOfCovering U f } :=
   ⟨U i, Opens.leSupr U i, i, rfl⟩
 #align Top.presheaf.presieve_of_covering.hom_of_index TopCat.Presheaf.presieveOfCovering.homOfIndex
+-/
 
+#print TopCat.Presheaf.presieveOfCovering.indexOfHom /-
 /-- By using the axiom of choice, a dependent pair `(V, f)` where `f : V ⟶ supr U` is a member of
 `presieve_of_covering U f` can be turned into an index `i : ι`, such that `V = U i`.
 -/
 def indexOfHom (f : Σ V, { f : V ⟶ iSup U // presieveOfCovering U f }) : ι :=
   f.2.2.some
 #align Top.presheaf.presieve_of_covering.index_of_hom TopCat.Presheaf.presieveOfCovering.indexOfHom
+-/
 
+#print TopCat.Presheaf.presieveOfCovering.indexOfHom_spec /-
 theorem indexOfHom_spec (f : Σ V, { f : V ⟶ iSup U // presieveOfCovering U f }) :
     f.1 = U (indexOfHom U f) :=
   f.2.2.choose_spec
 #align Top.presheaf.presieve_of_covering.index_of_hom_spec TopCat.Presheaf.presieveOfCovering.indexOfHom_spec
+-/
 
 end PresieveOfCovering
 
@@ -139,6 +159,7 @@ namespace TopCat.Opens
 
 variable {X : TopCat} {ι : Type _}
 
+#print TopCat.Opens.coverDense_iff_isBasis /-
 theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
     CoverDense (Opens.grothendieckTopology X) B ↔ Opens.IsBasis (Set.range B.obj) :=
   by
@@ -148,11 +169,14 @@ theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
   intro hb; constructor; intro U x hx; rcases hb hx with ⟨_, ⟨i, rfl⟩, hx, hi⟩
   exact ⟨B.obj i, ⟨⟨hi⟩⟩, ⟨⟨i, 𝟙 _, ⟨⟨hi⟩⟩, rfl⟩⟩, hx⟩
 #align Top.opens.cover_dense_iff_is_basis TopCat.Opens.coverDense_iff_isBasis
+-/
 
+#print TopCat.Opens.coverDense_inducedFunctor /-
 theorem coverDense_inducedFunctor {B : ι → Opens X} (h : Opens.IsBasis (Set.range B)) :
     CoverDense (Opens.grothendieckTopology X) (inducedFunctor B) :=
   (coverDense_iff_isBasis _).2 h
 #align Top.opens.cover_dense_induced_functor TopCat.Opens.coverDense_inducedFunctor
+-/
 
 end TopCat.Opens
 
@@ -164,6 +188,7 @@ variable {C : Type u} [Category.{v} C]
 
 variable {X Y : TopCat.{w}} {f : X ⟶ Y} {F : Y.Presheaf C}
 
+#print OpenEmbedding.compatiblePreserving /-
 theorem OpenEmbedding.compatiblePreserving (hf : OpenEmbedding f) :
     CompatiblePreserving (Opens.grothendieckTopology Y) hf.IsOpenMap.Functor :=
   by
@@ -175,7 +200,9 @@ theorem OpenEmbedding.compatiblePreserving (hf : OpenEmbedding f) :
   obtain ⟨_, _, rfl⟩ := i.le h
   exact ⟨_, rfl⟩
 #align open_embedding.compatible_preserving OpenEmbedding.compatiblePreserving
+-/
 
+#print IsOpenMap.coverPreserving /-
 theorem IsOpenMap.coverPreserving (hf : IsOpenMap f) :
     CoverPreserving (Opens.grothendieckTopology X) (Opens.grothendieckTopology Y) hf.Functor :=
   by
@@ -184,11 +211,14 @@ theorem IsOpenMap.coverPreserving (hf : IsOpenMap f) :
   obtain ⟨V, i, hV, hxV⟩ := hU x hx
   exact ⟨_, hf.functor.map i, ⟨_, i, 𝟙 _, hV, rfl⟩, Set.mem_image_of_mem f hxV⟩
 #align is_open_map.cover_preserving IsOpenMap.coverPreserving
+-/
 
+#print TopCat.Presheaf.isSheaf_of_openEmbedding /-
 theorem TopCat.Presheaf.isSheaf_of_openEmbedding (h : OpenEmbedding f) (hF : F.IsSheaf) :
     IsSheaf (h.IsOpenMap.Functor.op ⋙ F) :=
   pullback_isSheaf_of_coverPreserving h.CompatiblePreserving h.IsOpenMap.CoverPreserving ⟨_, hF⟩
 #align Top.presheaf.is_sheaf_of_open_embedding TopCat.Presheaf.isSheaf_of_openEmbedding
+-/
 
 end OpenEmbedding
 
@@ -202,15 +232,19 @@ variable {X : TopCat.{w}} {ι : Type _} {B : ι → Opens X}
 
 variable (F : X.Presheaf C) (F' : Sheaf C X) (h : Opens.IsBasis (Set.range B))
 
+#print TopCat.Sheaf.isTerminalOfEmpty /-
 /-- The empty component of a sheaf is terminal -/
 def isTerminalOfEmpty (F : Sheaf C X) : Limits.IsTerminal (F.val.obj (op ⊥)) :=
   F.isTerminalOfBotCover ⊥ (by tidy)
 #align Top.sheaf.is_terminal_of_empty TopCat.Sheaf.isTerminalOfEmpty
+-/
 
+#print TopCat.Sheaf.isTerminalOfEqEmpty /-
 /-- A variant of `is_terminal_of_empty` that is easier to `apply`. -/
 def isTerminalOfEqEmpty (F : X.Sheaf C) {U : Opens X} (h : U = ⊥) :
     Limits.IsTerminal (F.val.obj (op U)) := by convert F.is_terminal_of_empty
 #align Top.sheaf.is_terminal_of_eq_empty TopCat.Sheaf.isTerminalOfEqEmpty
+-/
 
 #print TopCat.Sheaf.restrictHomEquivHom /-
 /-- If a family `B` of open sets forms a basis of the topology on `X`, and if `F'`
@@ -222,17 +256,19 @@ def restrictHomEquivHom : ((inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op
 #align Top.sheaf.restrict_hom_equiv_hom TopCat.Sheaf.restrictHomEquivHom
 -/
 
+#print TopCat.Sheaf.extend_hom_app /-
 @[simp]
 theorem extend_hom_app (α : (inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.1) (i : ι) :
     (restrictHomEquivHom F F' h α).app (op (B i)) = α.app (op i) := by
   nth_rw 2 [← (restrict_hom_equiv_hom F F' h).left_inv α]; rfl
 #align Top.sheaf.extend_hom_app TopCat.Sheaf.extend_hom_app
+-/
 
-include h
-
+#print TopCat.Sheaf.hom_ext /-
 theorem hom_ext {α β : F ⟶ F'.1} (he : ∀ i, α.app (op (B i)) = β.app (op (B i))) : α = β := by
   apply (restrict_hom_equiv_hom F F' h).symm.Injective; ext i; exact he i.unop
 #align Top.sheaf.hom_ext TopCat.Sheaf.hom_ext
+-/
 
 end TopCat.Sheaf
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -43,12 +43,12 @@ variable {X : TopCat.{w}}
 /-- Given a presieve `R` on `U`, we obtain a covering family of open sets in `X`, by taking as index
 type the type of dependent pairs `(V, f)`, where `f : V ⟶ U` is in `R`.
 -/
-def coveringOfPresieve (U : Opens X) (R : Presieve U) : (ΣV, { f : V ⟶ U // R f }) → Opens X :=
+def coveringOfPresieve (U : Opens X) (R : Presieve U) : (Σ V, { f : V ⟶ U // R f }) → Opens X :=
   fun f => f.1
 #align Top.presheaf.covering_of_presieve TopCat.Presheaf.coveringOfPresieve
 
 @[simp]
-theorem coveringOfPresieve_apply (U : Opens X) (R : Presieve U) (f : ΣV, { f : V ⟶ U // R f }) :
+theorem coveringOfPresieve_apply (U : Opens X) (R : Presieve U) (f : Σ V, { f : V ⟶ U // R f }) :
     coveringOfPresieve U R f = f.1 :=
   rfl
 #align Top.presheaf.covering_of_presieve_apply TopCat.Presheaf.coveringOfPresieve_apply
@@ -115,18 +115,18 @@ theorem mem_grothendieckTopology :
 /-- An index `i : ι` can be turned into a dependent pair `(V, f)`, where `V` is an open set and
 `f : V ⟶ supr U` is a member of `presieve_of_covering U f`.
 -/
-def homOfIndex (i : ι) : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f } :=
+def homOfIndex (i : ι) : Σ V, { f : V ⟶ iSup U // presieveOfCovering U f } :=
   ⟨U i, Opens.leSupr U i, i, rfl⟩
 #align Top.presheaf.presieve_of_covering.hom_of_index TopCat.Presheaf.presieveOfCovering.homOfIndex
 
 /-- By using the axiom of choice, a dependent pair `(V, f)` where `f : V ⟶ supr U` is a member of
 `presieve_of_covering U f` can be turned into an index `i : ι`, such that `V = U i`.
 -/
-def indexOfHom (f : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f }) : ι :=
+def indexOfHom (f : Σ V, { f : V ⟶ iSup U // presieveOfCovering U f }) : ι :=
   f.2.2.some
 #align Top.presheaf.presieve_of_covering.index_of_hom TopCat.Presheaf.presieveOfCovering.indexOfHom
 
-theorem indexOfHom_spec (f : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f }) :
+theorem indexOfHom_spec (f : Σ V, { f : V ⟶ iSup U // presieveOfCovering U f }) :
     f.1 = U (indexOfHom U f) :=
   f.2.2.choose_spec
 #align Top.presheaf.presieve_of_covering.index_of_hom_spec TopCat.Presheaf.presieveOfCovering.indexOfHom_spec

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -40,12 +40,6 @@ namespace TopCat.Presheaf
 
 variable {X : TopCat.{w}}
 
-/- warning: Top.presheaf.covering_of_presieve -> TopCat.Presheaf.coveringOfPresieve is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align Top.presheaf.covering_of_presieve TopCat.Presheaf.coveringOfPresieveₓ'. -/
 /-- Given a presieve `R` on `U`, we obtain a covering family of open sets in `X`, by taking as index
 type the type of dependent pairs `(V, f)`, where `f : V ⟶ U` is in `R`.
 -/
@@ -53,9 +47,6 @@ def coveringOfPresieve (U : Opens X) (R : Presieve U) : (ΣV, { f : V ⟶ U // R
   fun f => f.1
 #align Top.presheaf.covering_of_presieve TopCat.Presheaf.coveringOfPresieve
 
-/- warning: Top.presheaf.covering_of_presieve_apply -> TopCat.Presheaf.coveringOfPresieve_apply is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align Top.presheaf.covering_of_presieve_apply TopCat.Presheaf.coveringOfPresieve_applyₓ'. -/
 @[simp]
 theorem coveringOfPresieve_apply (U : Opens X) (R : Presieve U) (f : ΣV, { f : V ⟶ U // R f }) :
     coveringOfPresieve U R f = f.1 :=
@@ -66,9 +57,6 @@ namespace CoveringOfPresieve
 
 variable (U : Opens X) (R : Presieve U)
 
-/- warning: Top.presheaf.covering_of_presieve.supr_eq_of_mem_grothendieck -> TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieck is a dubious translation:
-<too large>
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 /--
 If `R` is a presieve in the grothendieck topology on `opens X`, the covering family associated to
 `R` really is _covering_, i.e. the union of all open sets equals `U`.
@@ -87,12 +75,6 @@ theorem iSup_eq_of_mem_grothendieck (hR : Sieve.generate R ∈ Opens.grothendiec
 
 end CoveringOfPresieve
 
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 /-- Given a family of opens `U : ι → opens X` and any open `Y : opens X`, we obtain a presieve
 on `Y` by declaring that a morphism `f : V ⟶ Y` is a member of the presieve if and only if
 there exists an index `i : ι` such that `V = U i`.
@@ -101,20 +83,11 @@ def presieveOfCoveringAux {ι : Type v} (U : ι → Opens X) (Y : Opens X) : Pre
   ∃ i, V = U i
 #align Top.presheaf.presieve_of_covering_aux TopCat.Presheaf.presieveOfCoveringAux
 
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 /-- Take `Y` to be `supr U` and obtain a presieve over `supr U`. -/
 def presieveOfCovering {ι : Type v} (U : ι → Opens X) : Presieve (iSup U) :=
   presieveOfCoveringAux U (iSup U)
 #align Top.presheaf.presieve_of_covering TopCat.Presheaf.presieveOfCovering
 
-/- warning: Top.presheaf.covering_presieve_eq_self -> TopCat.Presheaf.covering_presieve_eq_self is a dubious translation:
-<too large>
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 /-- Given a presieve `R` on `Y`, if we take its associated family of opens via
     `covering_of_presieve` (which may not cover `Y` if `R` is not covering), and take
     the presieve on `Y` associated to the family of opens via `presieve_of_covering_aux`,
@@ -129,9 +102,6 @@ namespace PresieveOfCovering
 
 variable {ι : Type v} (U : ι → Opens X)
 
-/- warning: Top.presheaf.presieve_of_covering.mem_grothendieck_topology -> TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align Top.presheaf.presieve_of_covering.mem_grothendieck_topology TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopologyₓ'. -/
 /-- The sieve generated by `presieve_of_covering U` is a member of the grothendieck topology.
 -/
 theorem mem_grothendieckTopology :
@@ -142,12 +112,6 @@ theorem mem_grothendieckTopology :
   exact ⟨U i, opens.le_supr U i, ⟨U i, 𝟙 _, opens.le_supr U i, ⟨i, rfl⟩, category.id_comp _⟩, hxi⟩
 #align Top.presheaf.presieve_of_covering.mem_grothendieck_topology TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology
 
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-Case conversion may be inaccurate. Consider using '#align Top.presheaf.presieve_of_covering.hom_of_index TopCat.Presheaf.presieveOfCovering.homOfIndexₓ'. -/
 /-- An index `i : ι` can be turned into a dependent pair `(V, f)`, where `V` is an open set and
 `f : V ⟶ supr U` is a member of `presieve_of_covering U f`.
 -/
@@ -155,12 +119,6 @@ def homOfIndex (i : ι) : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f } :=
   ⟨U i, Opens.leSupr U i, i, rfl⟩
 #align Top.presheaf.presieve_of_covering.hom_of_index TopCat.Presheaf.presieveOfCovering.homOfIndex
 
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 /-- By using the axiom of choice, a dependent pair `(V, f)` where `f : V ⟶ supr U` is a member of
 `presieve_of_covering U f` can be turned into an index `i : ι`, such that `V = U i`.
 -/
@@ -168,9 +126,6 @@ def indexOfHom (f : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f }) : ι :=
   f.2.2.some
 #align Top.presheaf.presieve_of_covering.index_of_hom TopCat.Presheaf.presieveOfCovering.indexOfHom
 
-/- warning: Top.presheaf.presieve_of_covering.index_of_hom_spec -> TopCat.Presheaf.presieveOfCovering.indexOfHom_spec is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align Top.presheaf.presieve_of_covering.index_of_hom_spec TopCat.Presheaf.presieveOfCovering.indexOfHom_specₓ'. -/
 theorem indexOfHom_spec (f : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f }) :
     f.1 = U (indexOfHom U f) :=
   f.2.2.choose_spec
@@ -184,12 +139,6 @@ namespace TopCat.Opens
 
 variable {X : TopCat} {ι : Type _}
 
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-Case conversion may be inaccurate. Consider using '#align Top.opens.cover_dense_iff_is_basis TopCat.Opens.coverDense_iff_isBasisₓ'. -/
 theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
     CoverDense (Opens.grothendieckTopology X) B ↔ Opens.IsBasis (Set.range B.obj) :=
   by
@@ -200,12 +149,6 @@ theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
   exact ⟨B.obj i, ⟨⟨hi⟩⟩, ⟨⟨i, 𝟙 _, ⟨⟨hi⟩⟩, rfl⟩⟩, hx⟩
 #align Top.opens.cover_dense_iff_is_basis TopCat.Opens.coverDense_iff_isBasis
 
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-Case conversion may be inaccurate. Consider using '#align Top.opens.cover_dense_induced_functor TopCat.Opens.coverDense_inducedFunctorₓ'. -/
 theorem coverDense_inducedFunctor {B : ι → Opens X} (h : Opens.IsBasis (Set.range B)) :
     CoverDense (Opens.grothendieckTopology X) (inducedFunctor B) :=
   (coverDense_iff_isBasis _).2 h
@@ -221,9 +164,6 @@ variable {C : Type u} [Category.{v} C]
 
 variable {X Y : TopCat.{w}} {f : X ⟶ Y} {F : Y.Presheaf C}
 
-/- warning: open_embedding.compatible_preserving -> OpenEmbedding.compatiblePreserving is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align open_embedding.compatible_preserving OpenEmbedding.compatiblePreservingₓ'. -/
 theorem OpenEmbedding.compatiblePreserving (hf : OpenEmbedding f) :
     CompatiblePreserving (Opens.grothendieckTopology Y) hf.IsOpenMap.Functor :=
   by
@@ -236,12 +176,6 @@ theorem OpenEmbedding.compatiblePreserving (hf : OpenEmbedding f) :
   exact ⟨_, rfl⟩
 #align open_embedding.compatible_preserving OpenEmbedding.compatiblePreserving
 
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 theorem IsOpenMap.coverPreserving (hf : IsOpenMap f) :
     CoverPreserving (Opens.grothendieckTopology X) (Opens.grothendieckTopology Y) hf.Functor :=
   by
@@ -251,9 +185,6 @@ theorem IsOpenMap.coverPreserving (hf : IsOpenMap f) :
   exact ⟨_, hf.functor.map i, ⟨_, i, 𝟙 _, hV, rfl⟩, Set.mem_image_of_mem f hxV⟩
 #align is_open_map.cover_preserving IsOpenMap.coverPreserving
 
-/- warning: Top.presheaf.is_sheaf_of_open_embedding -> TopCat.Presheaf.isSheaf_of_openEmbedding is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align Top.presheaf.is_sheaf_of_open_embedding TopCat.Presheaf.isSheaf_of_openEmbeddingₓ'. -/
 theorem TopCat.Presheaf.isSheaf_of_openEmbedding (h : OpenEmbedding f) (hF : F.IsSheaf) :
     IsSheaf (h.IsOpenMap.Functor.op ⋙ F) :=
   pullback_isSheaf_of_coverPreserving h.CompatiblePreserving h.IsOpenMap.CoverPreserving ⟨_, hF⟩
@@ -271,23 +202,11 @@ variable {X : TopCat.{w}} {ι : Type _} {B : ι → Opens X}
 
 variable (F : X.Presheaf C) (F' : Sheaf C X) (h : Opens.IsBasis (Set.range B))
 
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-Case conversion may be inaccurate. Consider using '#align Top.sheaf.is_terminal_of_empty TopCat.Sheaf.isTerminalOfEmptyₓ'. -/
 /-- The empty component of a sheaf is terminal -/
 def isTerminalOfEmpty (F : Sheaf C X) : Limits.IsTerminal (F.val.obj (op ⊥)) :=
   F.isTerminalOfBotCover ⊥ (by tidy)
 #align Top.sheaf.is_terminal_of_empty TopCat.Sheaf.isTerminalOfEmpty
 
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-Case conversion may be inaccurate. Consider using '#align Top.sheaf.is_terminal_of_eq_empty TopCat.Sheaf.isTerminalOfEqEmptyₓ'. -/
 /-- A variant of `is_terminal_of_empty` that is easier to `apply`. -/
 def isTerminalOfEqEmpty (F : X.Sheaf C) {U : Opens X} (h : U = ⊥) :
     Limits.IsTerminal (F.val.obj (op U)) := by convert F.is_terminal_of_empty
@@ -303,9 +222,6 @@ def restrictHomEquivHom : ((inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op
 #align Top.sheaf.restrict_hom_equiv_hom TopCat.Sheaf.restrictHomEquivHom
 -/
 
-/- warning: Top.sheaf.extend_hom_app -> TopCat.Sheaf.extend_hom_app is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align Top.sheaf.extend_hom_app TopCat.Sheaf.extend_hom_appₓ'. -/
 @[simp]
 theorem extend_hom_app (α : (inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.1) (i : ι) :
     (restrictHomEquivHom F F' h α).app (op (B i)) = α.app (op i) := by
@@ -314,9 +230,6 @@ theorem extend_hom_app (α : (inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).
 
 include h
 
-/- warning: Top.sheaf.hom_ext -> TopCat.Sheaf.hom_ext is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align Top.sheaf.hom_ext TopCat.Sheaf.hom_extₓ'. -/
 theorem hom_ext {α β : F ⟶ F'.1} (he : ∀ i, α.app (op (B i)) = β.app (op (B i))) : α = β := by
   apply (restrict_hom_equiv_hom F F' h).symm.Injective; ext i; exact he i.unop
 #align Top.sheaf.hom_ext TopCat.Sheaf.hom_ext

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -121,9 +121,7 @@ Case conversion may be inaccurate. Consider using '#align Top.presheaf.covering_
     then we get back the original presieve `R`. -/
 @[simp]
 theorem covering_presieve_eq_self {Y : Opens X} (R : Presieve Y) :
-    presieveOfCoveringAux (coveringOfPresieve Y R) Y = R :=
-  by
-  ext (Z f)
+    presieveOfCoveringAux (coveringOfPresieve Y R) Y = R := by ext (Z f);
   exact ⟨fun ⟨⟨_, _, h⟩, rfl⟩ => by convert h, fun h => ⟨⟨Z, f, h⟩, rfl⟩⟩
 #align Top.presheaf.covering_presieve_eq_self TopCat.Presheaf.covering_presieve_eq_self
 
@@ -310,10 +308,8 @@ def restrictHomEquivHom : ((inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op
 Case conversion may be inaccurate. Consider using '#align Top.sheaf.extend_hom_app TopCat.Sheaf.extend_hom_appₓ'. -/
 @[simp]
 theorem extend_hom_app (α : (inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.1) (i : ι) :
-    (restrictHomEquivHom F F' h α).app (op (B i)) = α.app (op i) :=
-  by
-  nth_rw 2 [← (restrict_hom_equiv_hom F F' h).left_inv α]
-  rfl
+    (restrictHomEquivHom F F' h α).app (op (B i)) = α.app (op i) := by
+  nth_rw 2 [← (restrict_hom_equiv_hom F F' h).left_inv α]; rfl
 #align Top.sheaf.extend_hom_app TopCat.Sheaf.extend_hom_app
 
 include h
@@ -321,11 +317,8 @@ include h
 /- warning: Top.sheaf.hom_ext -> TopCat.Sheaf.hom_ext is a dubious translation:
 <too large>
 Case conversion may be inaccurate. Consider using '#align Top.sheaf.hom_ext TopCat.Sheaf.hom_extₓ'. -/
-theorem hom_ext {α β : F ⟶ F'.1} (he : ∀ i, α.app (op (B i)) = β.app (op (B i))) : α = β :=
-  by
-  apply (restrict_hom_equiv_hom F F' h).symm.Injective
-  ext i
-  exact he i.unop
+theorem hom_ext {α β : F ⟶ F'.1} (he : ∀ i, α.app (op (B i)) = β.app (op (B i))) : α = β := by
+  apply (restrict_hom_equiv_hom F F' h).symm.Injective; ext i; exact he i.unop
 #align Top.sheaf.hom_ext TopCat.Sheaf.hom_ext
 
 end TopCat.Sheaf

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -54,10 +54,7 @@ def coveringOfPresieve (U : Opens X) (R : Presieve U) : (ΣV, { f : V ⟶ U // R
 #align Top.presheaf.covering_of_presieve TopCat.Presheaf.coveringOfPresieve
 
 /- warning: Top.presheaf.covering_of_presieve_apply -> TopCat.Presheaf.coveringOfPresieve_apply is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align Top.presheaf.covering_of_presieve_apply TopCat.Presheaf.coveringOfPresieve_applyₓ'. -/
 @[simp]
 theorem coveringOfPresieve_apply (U : Opens X) (R : Presieve U) (f : ΣV, { f : V ⟶ U // R f }) :
@@ -70,10 +67,7 @@ namespace CoveringOfPresieve
 variable (U : Opens X) (R : Presieve U)
 
 /- warning: Top.presheaf.covering_of_presieve.supr_eq_of_mem_grothendieck -> TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieck is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align Top.presheaf.covering_of_presieve.supr_eq_of_mem_grothendieck TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieckₓ'. -/
 /--
 If `R` is a presieve in the grothendieck topology on `opens X`, the covering family associated to
@@ -119,10 +113,7 @@ def presieveOfCovering {ι : Type v} (U : ι → Opens X) : Presieve (iSup U) :=
 #align Top.presheaf.presieve_of_covering TopCat.Presheaf.presieveOfCovering
 
 /- warning: Top.presheaf.covering_presieve_eq_self -> TopCat.Presheaf.covering_presieve_eq_self is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align Top.presheaf.covering_presieve_eq_self TopCat.Presheaf.covering_presieve_eq_selfₓ'. -/
 /-- Given a presieve `R` on `Y`, if we take its associated family of opens via
     `covering_of_presieve` (which may not cover `Y` if `R` is not covering), and take
@@ -141,10 +132,7 @@ namespace PresieveOfCovering
 variable {ι : Type v} (U : ι → Opens X)
 
 /- warning: Top.presheaf.presieve_of_covering.mem_grothendieck_topology -> TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align Top.presheaf.presieve_of_covering.mem_grothendieck_topology TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopologyₓ'. -/
 /-- The sieve generated by `presieve_of_covering U` is a member of the grothendieck topology.
 -/
@@ -183,10 +171,7 @@ def indexOfHom (f : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f }) : ι :=
 #align Top.presheaf.presieve_of_covering.index_of_hom TopCat.Presheaf.presieveOfCovering.indexOfHom
 
 /- warning: Top.presheaf.presieve_of_covering.index_of_hom_spec -> TopCat.Presheaf.presieveOfCovering.indexOfHom_spec is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align Top.presheaf.presieve_of_covering.index_of_hom_spec TopCat.Presheaf.presieveOfCovering.indexOfHom_specₓ'. -/
 theorem indexOfHom_spec (f : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f }) :
     f.1 = U (indexOfHom U f) :=
@@ -239,10 +224,7 @@ variable {C : Type u} [Category.{v} C]
 variable {X Y : TopCat.{w}} {f : X ⟶ Y} {F : Y.Presheaf C}
 
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 Case conversion may be inaccurate. Consider using '#align open_embedding.compatible_preserving OpenEmbedding.compatiblePreservingₓ'. -/
 theorem OpenEmbedding.compatiblePreserving (hf : OpenEmbedding f) :
     CompatiblePreserving (Opens.grothendieckTopology Y) hf.IsOpenMap.Functor :=
@@ -272,10 +254,7 @@ theorem IsOpenMap.coverPreserving (hf : IsOpenMap f) :
 #align is_open_map.cover_preserving IsOpenMap.coverPreserving
 
 /- warning: Top.presheaf.is_sheaf_of_open_embedding -> TopCat.Presheaf.isSheaf_of_openEmbedding is a dubious translation:
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 Case conversion may be inaccurate. Consider using '#align Top.presheaf.is_sheaf_of_open_embedding TopCat.Presheaf.isSheaf_of_openEmbeddingₓ'. -/
 theorem TopCat.Presheaf.isSheaf_of_openEmbedding (h : OpenEmbedding f) (hF : F.IsSheaf) :
     IsSheaf (h.IsOpenMap.Functor.op ⋙ F) :=
@@ -327,10 +306,7 @@ def restrictHomEquivHom : ((inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op
 -/
 
 /- warning: Top.sheaf.extend_hom_app -> TopCat.Sheaf.extend_hom_app is a dubious translation:
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(CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, 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(CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} 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(Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) (Opposite.{succ u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))) (CategoryTheory.Category.opposite.{u2, u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} 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(CategoryTheory.inducedFunctor.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) (CategoryTheory.Sheaf.val.{u2, u3, u2, u4} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) (Opens.grothendieckTopology.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) C _inst_1 F'))) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : Quiver.Hom.{max (succ u3) (succ u1), max (max (max u4 u1) u3) u2} (CategoryTheory.Functor.{u2, u3, u1, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} 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(TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) C _inst_1))) (CategoryTheory.Functor.comp.{u2, u2, u3, u1, u2, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) 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X))) (CategoryTheory.Category.opposite.{u2, u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B) (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) 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X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) (Opposite.{succ u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))) (CategoryTheory.Category.opposite.{u2, u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B) (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) (CategoryTheory.inducedFunctor.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) 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(Opens.grothendieckTopology.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) C _inst_1 F'))) => Quiver.Hom.{max (succ u3) (succ u2), max (max u4 u3) u2} (TopCat.Presheaf.{u2, u3, u4} C _inst_1 X) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u2, max (max u4 u3) u2} (TopCat.Presheaf.{u2, u3, u4} C _inst_1 X) (CategoryTheory.Category.toCategoryStruct.{max u3 u2, max (max u4 u3) u2} (TopCat.Presheaf.{u2, u3, u4} C _inst_1 X) (TopCat.instCategoryPresheaf.{u2, u3, u4} C _inst_1 X))) F (CategoryTheory.Sheaf.val.{u2, u3, u2, u4} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) 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(CategoryTheory.Category.toCategoryStruct.{max u3 u2, max (max u4 u3) u2} (TopCat.Presheaf.{u2, u3, u4} C _inst_1 X) (TopCat.instCategoryPresheaf.{u2, u3, u4} C _inst_1 X))) F (CategoryTheory.Sheaf.val.{u2, u3, u2, u4} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) (Opens.grothendieckTopology.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) C _inst_1 F'))) (TopCat.Sheaf.restrictHomEquivHom.{u2, u3, u4, u1} C _inst_1 X ι B F F' h) α) (Opposite.op.{succ u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (B i))) (CategoryTheory.NatTrans.app.{u2, u3, u1, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) C _inst_1 (CategoryTheory.Functor.comp.{u2, u2, u3, u1, u2, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) (Opposite.{succ u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))) (CategoryTheory.Category.opposite.{u2, u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B) (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) (CategoryTheory.inducedFunctor.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) F) (CategoryTheory.Functor.comp.{u2, u2, u3, u1, u2, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) (Opposite.{succ u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))) (CategoryTheory.Category.opposite.{u2, u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} 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+<too large>
 Case conversion may be inaccurate. Consider using '#align Top.sheaf.extend_hom_app TopCat.Sheaf.extend_hom_appₓ'. -/
 @[simp]
 theorem extend_hom_app (α : (inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.1) (i : ι) :
@@ -343,10 +319,7 @@ theorem extend_hom_app (α : (inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).
 include h
 
 /- warning: Top.sheaf.hom_ext -> TopCat.Sheaf.hom_ext is a dubious translation:
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(CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))))))) C _inst_1 F (CategoryTheory.Sheaf.val.{u1, u2, u1, u3} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (Opens.grothendieckTopology.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) C _inst_1 F') β (Opposite.op.{succ u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (B i)))) -> (Eq.{max (succ u2) (succ u1)} (Quiver.Hom.{max (succ u2) (succ u1), max (max u3 u2) u1} (TopCat.Presheaf.{u1, u2, u3} C _inst_1 X) (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (max u3 u2) u1} (TopCat.Presheaf.{u1, u2, u3} C _inst_1 X) (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (max u3 u2) u1} (TopCat.Presheaf.{u1, u2, u3} C _inst_1 X) (TopCat.instCategoryPresheaf.{u1, u2, u3} C _inst_1 X))) F (CategoryTheory.Sheaf.val.{u1, u2, u1, u3} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (Opens.grothendieckTopology.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) C _inst_1 F')) α β))
+<too large>
 Case conversion may be inaccurate. Consider using '#align Top.sheaf.hom_ext TopCat.Sheaf.hom_extₓ'. -/
 theorem hom_ext {α β : F ⟶ F'.1} (he : ∀ i, α.app (op (B i)) = β.app (op (B i))) : α = β :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Justus Springer
 
 ! This file was ported from Lean 3 source module topology.sheaves.sheaf_condition.sites
-! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc
+! leanprover-community/mathlib commit 4280f5f32e16755ec7985ce11e189b6cd6ff6735
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.CategoryTheory.Sites.DenseSubsite
 
 # Coverings and sieves; from sheaves on sites and sheaves on spaces
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file, we connect coverings in a topological space to sieves in the associated Grothendieck
 topology, in preparation of connecting the sheaf condition on sites to the various sheaf conditions
 on spaces.

mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952

@@ -37,6 +37,12 @@ namespace TopCat.Presheaf
 
 variable {X : TopCat.{w}}
 
+/- warning: Top.presheaf.covering_of_presieve -> TopCat.Presheaf.coveringOfPresieve is a dubious translation:
+lean 3 declaration is
+  forall {X : TopCat.{u1}} (U : TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (R : CategoryTheory.Presieve.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))) U), (Sigma.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (fun (V : TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) => Subtype.{succ u1} (Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))))) V U) (fun (f : Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))))) V U) => R V f))) -> (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))
+but is expected to have type
+  forall {X : TopCat.{u1}} (U : TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (R : CategoryTheory.Presieve.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) U), (Sigma.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (fun (V : TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) => Subtype.{succ u1} (Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V U) (fun (f : Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V U) => R V f))) -> (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))
+Case conversion may be inaccurate. Consider using '#align Top.presheaf.covering_of_presieve TopCat.Presheaf.coveringOfPresieveₓ'. -/
 /-- Given a presieve `R` on `U`, we obtain a covering family of open sets in `X`, by taking as index
 type the type of dependent pairs `(V, f)`, where `f : V ⟶ U` is in `R`.
 -/
@@ -44,6 +50,12 @@ def coveringOfPresieve (U : Opens X) (R : Presieve U) : (ΣV, { f : V ⟶ U // R
   fun f => f.1
 #align Top.presheaf.covering_of_presieve TopCat.Presheaf.coveringOfPresieve
 
+/- warning: Top.presheaf.covering_of_presieve_apply -> TopCat.Presheaf.coveringOfPresieve_apply is a dubious translation:
+lean 3 declaration is
+  forall {X : TopCat.{u1}} (U : TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (R : CategoryTheory.Presieve.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} 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(TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))))) V U) (fun (f : 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+but is expected to have type
+  forall {X : TopCat.{u1}} (U : TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (R : CategoryTheory.Presieve.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) U) (f : Sigma.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (fun (V : TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) => Subtype.{succ u1} (Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} 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(TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V U) => R V f))), Eq.{succ u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopCat.Presheaf.coveringOfPresieve.{u1} X U R f) (Sigma.fst.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (fun (V : TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) => Subtype.{succ u1} (Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V U) (fun (f : Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V U) => R V f)) f)
+Case conversion may be inaccurate. Consider using '#align Top.presheaf.covering_of_presieve_apply TopCat.Presheaf.coveringOfPresieve_applyₓ'. -/
 @[simp]
 theorem coveringOfPresieve_apply (U : Opens X) (R : Presieve U) (f : ΣV, { f : V ⟶ U // R f }) :
     coveringOfPresieve U R f = f.1 :=
@@ -54,6 +66,12 @@ namespace CoveringOfPresieve
 
 variable (U : Opens X) (R : Presieve U)
 
+/- warning: Top.presheaf.covering_of_presieve.supr_eq_of_mem_grothendieck -> TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieck is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) U)) (CategoryTheory.Sieve.generate.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) U R) (CategoryTheory.GrothendieckTopology.sieves.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (Opens.grothendieckTopology.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) U)) -> (Eq.{succ u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (iSup.{u1, succ u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))) (Sigma.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (fun (V : TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) => Subtype.{succ u1} (Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V U) (fun (f : Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V U) => R V f))) (TopCat.Presheaf.coveringOfPresieve.{u1} X U R)) U)
+Case conversion may be inaccurate. Consider using '#align Top.presheaf.covering_of_presieve.supr_eq_of_mem_grothendieck TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieckₓ'. -/
 /--
 If `R` is a presieve in the grothendieck topology on `opens X`, the covering family associated to
 `R` really is _covering_, i.e. the union of all open sets equals `U`.
@@ -72,6 +90,12 @@ theorem iSup_eq_of_mem_grothendieck (hR : Sieve.generate R ∈ Opens.grothendiec
 
 end CoveringOfPresieve
 
+/- warning: Top.presheaf.presieve_of_covering_aux -> TopCat.Presheaf.presieveOfCoveringAux is a dubious translation:
+lean 3 declaration is
+  forall {X : TopCat.{u1}} {ι : Type.{u2}}, (ι -> (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))) -> (forall (Y : TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)), CategoryTheory.Presieve.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))) Y)
+but is expected to have type
+  forall {X : TopCat.{u1}} {ι : Type.{u2}}, (ι -> (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))) -> (forall (Y : TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)), CategoryTheory.Presieve.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) Y)
+Case conversion may be inaccurate. Consider using '#align Top.presheaf.presieve_of_covering_aux TopCat.Presheaf.presieveOfCoveringAuxₓ'. -/
 /-- Given a family of opens `U : ι → opens X` and any open `Y : opens X`, we obtain a presieve
 on `Y` by declaring that a morphism `f : V ⟶ Y` is a member of the presieve if and only if
 there exists an index `i : ι` such that `V = U i`.
@@ -80,11 +104,23 @@ def presieveOfCoveringAux {ι : Type v} (U : ι → Opens X) (Y : Opens X) : Pre
   ∃ i, V = U i
 #align Top.presheaf.presieve_of_covering_aux TopCat.Presheaf.presieveOfCoveringAux
 
+/- warning: Top.presheaf.presieve_of_covering -> TopCat.Presheaf.presieveOfCovering is a dubious translation:
+lean 3 declaration is
+  forall {X : TopCat.{u1}} {ι : Type.{u2}} (U : ι -> (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))), CategoryTheory.Presieve.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (TopologicalSpace.Opens.completeLattice.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)))) ι U)
+but is expected to have type
+  forall {X : TopCat.{u1}} {ι : Type.{u2}} (U : ι -> (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))), CategoryTheory.Presieve.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))) ι U)
+Case conversion may be inaccurate. Consider using '#align Top.presheaf.presieve_of_covering TopCat.Presheaf.presieveOfCoveringₓ'. -/
 /-- Take `Y` to be `supr U` and obtain a presieve over `supr U`. -/
 def presieveOfCovering {ι : Type v} (U : ι → Opens X) : Presieve (iSup U) :=
   presieveOfCoveringAux U (iSup U)
 #align Top.presheaf.presieve_of_covering TopCat.Presheaf.presieveOfCovering
 
+/- warning: Top.presheaf.covering_presieve_eq_self -> TopCat.Presheaf.covering_presieve_eq_self is a dubious translation:
+lean 3 declaration is
+  forall {X : TopCat.{u1}} {Y : TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)} (R : CategoryTheory.Presieve.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} 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+but is expected to have type
+  forall {X : TopCat.{u1}} {Y : TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)} (R : CategoryTheory.Presieve.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) Y), Eq.{succ u1} (CategoryTheory.Presieve.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) 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(TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V Y) => R V f))) (TopCat.Presheaf.coveringOfPresieve.{u1} X Y R) Y) R
+Case conversion may be inaccurate. Consider using '#align Top.presheaf.covering_presieve_eq_self TopCat.Presheaf.covering_presieve_eq_selfₓ'. -/
 /-- Given a presieve `R` on `Y`, if we take its associated family of opens via
     `covering_of_presieve` (which may not cover `Y` if `R` is not covering), and take
     the presieve on `Y` associated to the family of opens via `presieve_of_covering_aux`,
@@ -101,6 +137,12 @@ namespace PresieveOfCovering
 
 variable {ι : Type v} (U : ι → Opens X)
 
+/- warning: Top.presheaf.presieve_of_covering.mem_grothendieck_topology -> TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {X : TopCat.{u1}} {ι : Type.{u2}} (U : ι -> (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))), Membership.mem.{u1, u1} (CategoryTheory.Sieve.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))) ι U)) (Set.{u1} (CategoryTheory.Sieve.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) 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(TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))) ι U))) (CategoryTheory.Sieve.generate.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))) ι U) (TopCat.Presheaf.presieveOfCovering.{u1, u2} X ι U)) (CategoryTheory.GrothendieckTopology.sieves.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (Opens.grothendieckTopology.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))) ι U))
+Case conversion may be inaccurate. Consider using '#align Top.presheaf.presieve_of_covering.mem_grothendieck_topology TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopologyₓ'. -/
 /-- The sieve generated by `presieve_of_covering U` is a member of the grothendieck topology.
 -/
 theorem mem_grothendieckTopology :
@@ -111,6 +153,12 @@ theorem mem_grothendieckTopology :
   exact ⟨U i, opens.le_supr U i, ⟨U i, 𝟙 _, opens.le_supr U i, ⟨i, rfl⟩, category.id_comp _⟩, hxi⟩
 #align Top.presheaf.presieve_of_covering.mem_grothendieck_topology TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology
 
+/- warning: Top.presheaf.presieve_of_covering.hom_of_index -> TopCat.Presheaf.presieveOfCovering.homOfIndex is a dubious translation:
+lean 3 declaration is
+  forall {X : TopCat.{u1}} {ι : Type.{u2}} (U : ι -> (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))), ι -> (Sigma.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (fun (V : TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) => Subtype.{succ u1} (Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))))) V (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (TopologicalSpace.Opens.completeLattice.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)))) ι U)) (fun (f : Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))))) V (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (ConditionallyCompleteLattice.toHasSup.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (TopologicalSpace.Opens.completeLattice.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)))) ι U)) => TopCat.Presheaf.presieveOfCovering.{u1, u2} X ι U V f)))
+but is expected to have type
+  forall {X : TopCat.{u1}} {ι : Type.{u2}} (U : ι -> (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))), ι -> (Sigma.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (fun (V : TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) => Subtype.{succ u1} (Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))) ι U)) (fun (f : Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))) ι U)) => TopCat.Presheaf.presieveOfCovering.{u1, u2} X ι U V f)))
+Case conversion may be inaccurate. Consider using '#align Top.presheaf.presieve_of_covering.hom_of_index TopCat.Presheaf.presieveOfCovering.homOfIndexₓ'. -/
 /-- An index `i : ι` can be turned into a dependent pair `(V, f)`, where `V` is an open set and
 `f : V ⟶ supr U` is a member of `presieve_of_covering U f`.
 -/
@@ -118,6 +166,12 @@ def homOfIndex (i : ι) : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f } :=
   ⟨U i, Opens.leSupr U i, i, rfl⟩
 #align Top.presheaf.presieve_of_covering.hom_of_index TopCat.Presheaf.presieveOfCovering.homOfIndex
 
+/- warning: Top.presheaf.presieve_of_covering.index_of_hom -> TopCat.Presheaf.presieveOfCovering.indexOfHom is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {X : TopCat.{u1}} {ι : Type.{u2}} (U : ι -> (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))), (Sigma.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (fun (V : TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) => Subtype.{succ u1} (Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))) ι U)) (fun (f : Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) V (iSup.{u1, succ u2} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (ConditionallyCompleteLattice.toSupSet.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))) ι U)) => TopCat.Presheaf.presieveOfCovering.{u1, u2} X ι U V f))) -> ι
+Case conversion may be inaccurate. Consider using '#align Top.presheaf.presieve_of_covering.index_of_hom TopCat.Presheaf.presieveOfCovering.indexOfHomₓ'. -/
 /-- By using the axiom of choice, a dependent pair `(V, f)` where `f : V ⟶ supr U` is a member of
 `presieve_of_covering U f` can be turned into an index `i : ι`, such that `V = U i`.
 -/
@@ -125,6 +179,12 @@ def indexOfHom (f : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f }) : ι :=
   f.2.2.some
 #align Top.presheaf.presieve_of_covering.index_of_hom TopCat.Presheaf.presieveOfCovering.indexOfHom
 
+/- warning: Top.presheaf.presieve_of_covering.index_of_hom_spec -> TopCat.Presheaf.presieveOfCovering.indexOfHom_spec is a dubious translation:
+lean 3 declaration is
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TopCat.Presheaf.presieveOfCovering.{u1, u2} X ι U V f)) f) (U (TopCat.Presheaf.presieveOfCovering.indexOfHom.{u1, u2} X ι U f))
+but is expected to have type
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(TopCat.topologicalSpace_coe.{u1} X)))) ι U)) (fun (f : Quiver.Hom.{succ u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} 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TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))) ι U)) => TopCat.Presheaf.presieveOfCovering.{u1, u2} X ι U V f)) f) (U (TopCat.Presheaf.presieveOfCovering.indexOfHom.{u1, u2} X ι U f))
+Case conversion may be inaccurate. Consider using '#align Top.presheaf.presieve_of_covering.index_of_hom_spec TopCat.Presheaf.presieveOfCovering.indexOfHom_specₓ'. -/
 theorem indexOfHom_spec (f : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f }) :
     f.1 = U (indexOfHom U f) :=
   f.2.2.choose_spec
@@ -138,6 +198,12 @@ namespace TopCat.Opens
 
 variable {X : TopCat} {ι : Type _}
 
+/- warning: Top.opens.cover_dense_iff_is_basis -> TopCat.Opens.coverDense_iff_isBasis is a dubious translation:
+lean 3 declaration is
+  forall {X : TopCat.{u1}} {ι : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u3, u2} ι] (B : CategoryTheory.Functor.{u3, u1, u2, u1} ι _inst_1 (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)))))), Iff (CategoryTheory.CoverDense.{u2, u3, u1, u1} ι _inst_1 (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))) (Opens.grothendieckTopology.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) B) (TopologicalSpace.Opens.IsBasis.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X) (Set.range.{u1, succ u2} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) ι (CategoryTheory.Functor.obj.{u3, u1, u2, u1} ι _inst_1 (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))) B)))
+but is expected to have type
+  forall {X : TopCat.{u1}} {ι : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u3, u2} ι] (B : CategoryTheory.Functor.{u3, u1, u2, u1} ι _inst_1 (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))))))), Iff (CategoryTheory.CoverDense.{u2, u3, u1, u1} ι _inst_1 (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (Opens.grothendieckTopology.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) B) (TopologicalSpace.Opens.IsBasis.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X) (Set.range.{u1, succ u2} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) ι (Prefunctor.obj.{succ u3, succ u1, u2, u1} ι (CategoryTheory.CategoryStruct.toQuiver.{u3, u2} ι (CategoryTheory.Category.toCategoryStruct.{u3, u2} ι _inst_1)) (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CategoryTheory.Category.toCategoryStruct.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))))) (CategoryTheory.Functor.toPrefunctor.{u3, u1, u2, u1} ι _inst_1 (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) B))))
+Case conversion may be inaccurate. Consider using '#align Top.opens.cover_dense_iff_is_basis TopCat.Opens.coverDense_iff_isBasisₓ'. -/
 theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
     CoverDense (Opens.grothendieckTopology X) B ↔ Opens.IsBasis (Set.range B.obj) :=
   by
@@ -148,6 +214,12 @@ theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
   exact ⟨B.obj i, ⟨⟨hi⟩⟩, ⟨⟨i, 𝟙 _, ⟨⟨hi⟩⟩, rfl⟩⟩, hx⟩
 #align Top.opens.cover_dense_iff_is_basis TopCat.Opens.coverDense_iff_isBasis
 
+/- warning: Top.opens.cover_dense_induced_functor -> TopCat.Opens.coverDense_inducedFunctor is a dubious translation:
+lean 3 declaration is
+  forall {X : TopCat.{u1}} {ι : Type.{u2}} {B : ι -> (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))}, (TopologicalSpace.Opens.IsBasis.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X) (Set.range.{u1, succ u2} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) ι B)) -> (CategoryTheory.CoverDense.{u2, u1, u1, u1} (CategoryTheory.InducedCategory.{u1, u2, u1} ι (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))) B) (CategoryTheory.InducedCategory.category.{u1, u2, u1} ι (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))) B) (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))) (Opens.grothendieckTopology.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CategoryTheory.inducedFunctor.{u1, u2, u1} ι (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))) B))
+but is expected to have type
+  forall {X : TopCat.{u2}} {ι : Type.{u1}} {B : ι -> (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))}, (TopologicalSpace.Opens.IsBasis.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X) (Set.range.{u2, succ u1} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) ι B)) -> (CategoryTheory.CoverDense.{u1, u2, u2, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B) (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) (Opens.grothendieckTopology.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CategoryTheory.inducedFunctor.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B))
+Case conversion may be inaccurate. Consider using '#align Top.opens.cover_dense_induced_functor TopCat.Opens.coverDense_inducedFunctorₓ'. -/
 theorem coverDense_inducedFunctor {B : ι → Opens X} (h : Opens.IsBasis (Set.range B)) :
     CoverDense (Opens.grothendieckTopology X) (inducedFunctor B) :=
   (coverDense_iff_isBasis _).2 h
@@ -163,6 +235,12 @@ variable {C : Type u} [Category.{v} C]
 
 variable {X Y : TopCat.{w}} {f : X ⟶ Y} {F : Y.Presheaf C}
 
+/- warning: open_embedding.compatible_preserving -> OpenEmbedding.compatiblePreserving is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align open_embedding.compatible_preserving OpenEmbedding.compatiblePreservingₓ'. -/
 theorem OpenEmbedding.compatiblePreserving (hf : OpenEmbedding f) :
     CompatiblePreserving (Opens.grothendieckTopology Y) hf.IsOpenMap.Functor :=
   by
@@ -175,6 +253,12 @@ theorem OpenEmbedding.compatiblePreserving (hf : OpenEmbedding f) :
   exact ⟨_, rfl⟩
 #align open_embedding.compatible_preserving OpenEmbedding.compatiblePreserving
 
+/- warning: is_open_map.cover_preserving -> IsOpenMap.coverPreserving is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align is_open_map.cover_preserving IsOpenMap.coverPreservingₓ'. -/
 theorem IsOpenMap.coverPreserving (hf : IsOpenMap f) :
     CoverPreserving (Opens.grothendieckTopology X) (Opens.grothendieckTopology Y) hf.Functor :=
   by
@@ -184,6 +268,12 @@ theorem IsOpenMap.coverPreserving (hf : IsOpenMap f) :
   exact ⟨_, hf.functor.map i, ⟨_, i, 𝟙 _, hV, rfl⟩, Set.mem_image_of_mem f hxV⟩
 #align is_open_map.cover_preserving IsOpenMap.coverPreserving
 
+/- warning: Top.presheaf.is_sheaf_of_open_embedding -> TopCat.Presheaf.isSheaf_of_openEmbedding is a dubious translation:
+lean 3 declaration is
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F))
+Case conversion may be inaccurate. Consider using '#align Top.presheaf.is_sheaf_of_open_embedding TopCat.Presheaf.isSheaf_of_openEmbeddingₓ'. -/
 theorem TopCat.Presheaf.isSheaf_of_openEmbedding (h : OpenEmbedding f) (hF : F.IsSheaf) :
     IsSheaf (h.IsOpenMap.Functor.op ⋙ F) :=
   pullback_isSheaf_of_coverPreserving h.CompatiblePreserving h.IsOpenMap.CoverPreserving ⟨_, hF⟩
@@ -201,16 +291,29 @@ variable {X : TopCat.{w}} {ι : Type _} {B : ι → Opens X}
 
 variable (F : X.Presheaf C) (F' : Sheaf C X) (h : Opens.IsBasis (Set.range B))
 
+/- warning: Top.sheaf.is_terminal_of_empty -> TopCat.Sheaf.isTerminalOfEmpty is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align Top.sheaf.is_terminal_of_empty TopCat.Sheaf.isTerminalOfEmptyₓ'. -/
 /-- The empty component of a sheaf is terminal -/
 def isTerminalOfEmpty (F : Sheaf C X) : Limits.IsTerminal (F.val.obj (op ⊥)) :=
   F.isTerminalOfBotCover ⊥ (by tidy)
 #align Top.sheaf.is_terminal_of_empty TopCat.Sheaf.isTerminalOfEmpty
 
+/- warning: Top.sheaf.is_terminal_of_eq_empty -> TopCat.Sheaf.isTerminalOfEqEmpty is a dubious translation:
+lean 3 declaration is
+  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] {X : TopCat.{u1}} (F : TopCat.Sheaf.{u1, u2, u3} C _inst_1 X) {U : TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) U (Bot.bot.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (CompleteLattice.toHasBot.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (TopologicalSpace.Opens.completeLattice.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))) -> (CategoryTheory.Limits.IsTerminal.{u2, u3} C _inst_1 (CategoryTheory.Functor.obj.{u1, u2, u1, u3} (Opposite.{succ u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))) (CategoryTheory.Category.opposite.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)))))) C _inst_1 (CategoryTheory.Sheaf.val.{u1, u2, u1, u3} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (SetLike.partialOrder.{u1, u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopologicalSpace.Opens.setLike.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X))))) (Opens.grothendieckTopology.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) C _inst_1 F) (Opposite.op.{succ u1} (TopologicalSpace.Opens.{u1} (coeSort.{succ (succ u1), succ (succ u1)} TopCat.{u1} Type.{u1} TopCat.hasCoeToSort.{u1} X) (TopCat.topologicalSpace.{u1} X)) U)))
+but is expected to have type
+  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] {X : TopCat.{u1}} (F : TopCat.Sheaf.{u1, u2, u3} C _inst_1 X) {U : TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)}, (Eq.{succ u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) U (Bot.bot.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toBot.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))))) -> (CategoryTheory.Limits.IsTerminal.{u2, u3} C _inst_1 (Prefunctor.obj.{succ u1, succ u2, u1, u3} (Opposite.{succ u1} 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(TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))))))) C _inst_1 (CategoryTheory.Sheaf.val.{u1, u2, u1, u3} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (Opens.grothendieckTopology.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) C _inst_1 F)) (Opposite.op.{succ u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) U)))
+Case conversion may be inaccurate. Consider using '#align Top.sheaf.is_terminal_of_eq_empty TopCat.Sheaf.isTerminalOfEqEmptyₓ'. -/
 /-- A variant of `is_terminal_of_empty` that is easier to `apply`. -/
 def isTerminalOfEqEmpty (F : X.Sheaf C) {U : Opens X} (h : U = ⊥) :
     Limits.IsTerminal (F.val.obj (op U)) := by convert F.is_terminal_of_empty
 #align Top.sheaf.is_terminal_of_eq_empty TopCat.Sheaf.isTerminalOfEqEmpty
 
+#print TopCat.Sheaf.restrictHomEquivHom /-
 /-- If a family `B` of open sets forms a basis of the topology on `X`, and if `F'`
     is a sheaf on `X`, then a homomorphism between a presheaf `F` on `X` and `F'`
     is equivalent to a homomorphism between their restrictions to the indexing type
@@ -218,7 +321,14 @@ def isTerminalOfEqEmpty (F : X.Sheaf C) {U : Opens X} (h : U = ⊥) :
 def restrictHomEquivHom : ((inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.1) ≃ (F ⟶ F'.1) :=
   @CoverDense.restrictHomEquivHom _ _ _ _ _ _ _ _ (Opens.coverDense_inducedFunctor h) _ F F'
 #align Top.sheaf.restrict_hom_equiv_hom TopCat.Sheaf.restrictHomEquivHom
+-/
 
+/- warning: Top.sheaf.extend_hom_app -> TopCat.Sheaf.extend_hom_app is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {X : TopCat.{u2}} {ι : Type.{u1}} {B : ι -> (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))} (F : TopCat.Presheaf.{u2, u3, u4} C _inst_1 X) (F' : TopCat.Sheaf.{u2, u3, u4} C _inst_1 X) (h : TopologicalSpace.Opens.IsBasis.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X) (Set.range.{u2, succ u1} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) ι B)) (α : Quiver.Hom.{max (succ u3) (succ u1), max (max (max u4 u1) u3) u2} (CategoryTheory.Functor.{u2, u3, u1, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u1, max (max (max u4 u3) u2) u1} (CategoryTheory.Functor.{u2, u3, u1, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u3 u1, max (max (max u4 u3) u2) u1} (CategoryTheory.Functor.{u2, u3, u1, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) C _inst_1) (CategoryTheory.Functor.category.{u2, u3, u1, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) C _inst_1))) (CategoryTheory.Functor.comp.{u2, u2, u3, u1, u2, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) (Opposite.{succ u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))) (CategoryTheory.Category.opposite.{u2, u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, 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(TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) (CategoryTheory.inducedFunctor.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) F) (CategoryTheory.Functor.comp.{u2, u2, u3, u1, u2, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) (Opposite.{succ u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))) (CategoryTheory.Category.opposite.{u2, u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} 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X))) (CategoryTheory.Category.opposite.{u2, u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} 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(Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B) (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) 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(CategoryTheory.Sheaf.val.{u2, u3, u2, u4} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) (Opens.grothendieckTopology.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) C _inst_1 F'))) (fun (_x : Quiver.Hom.{max (succ u3) (succ u1), max (max (max u4 u1) u3) u2} (CategoryTheory.Functor.{u2, u3, u1, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) 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(CategoryTheory.Category.opposite.{u2, u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} 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(Opens.grothendieckTopology.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) C _inst_1 F'))) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : Quiver.Hom.{max (succ u3) (succ u1), max (max (max u4 u1) u3) u2} (CategoryTheory.Functor.{u2, u3, u1, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} 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(CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} 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(CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) (Opposite.{succ u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))) (CategoryTheory.Category.opposite.{u2, u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} 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(CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) (CategoryTheory.inducedFunctor.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) (CategoryTheory.Sheaf.val.{u2, u3, u2, u4} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) (Opens.grothendieckTopology.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) C _inst_1 F'))) (Quiver.Hom.{max (succ u3) (succ u2), max (max u4 u3) u2} (TopCat.Presheaf.{u2, u3, u4} C _inst_1 X) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u2, max (max u4 u3) u2} (TopCat.Presheaf.{u2, u3, u4} C _inst_1 X) (CategoryTheory.Category.toCategoryStruct.{max u3 u2, max (max u4 u3) u2} (TopCat.Presheaf.{u2, u3, u4} C _inst_1 X) (TopCat.instCategoryPresheaf.{u2, u3, u4} C _inst_1 X))) F (CategoryTheory.Sheaf.val.{u2, u3, u2, u4} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) (Opens.grothendieckTopology.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) C _inst_1 F'))) (TopCat.Sheaf.restrictHomEquivHom.{u2, u3, u4, u1} C _inst_1 X ι B F F' h) α) (Opposite.op.{succ u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (B i))) (CategoryTheory.NatTrans.app.{u2, u3, u1, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) C _inst_1 (CategoryTheory.Functor.comp.{u2, u2, u3, u1, u2, u4} (Opposite.{succ u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B)) (CategoryTheory.Category.opposite.{u2, u1} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) B)) (Opposite.{succ u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))) (CategoryTheory.Category.opposite.{u2, u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X))))))) C _inst_1 (CategoryTheory.Functor.op.{u2, u2, u1, u2} (CategoryTheory.InducedCategory.{u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) B) (CategoryTheory.InducedCategory.category.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} 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X) (TopCat.topologicalSpace_coe.{u2} X)) (Preorder.smallCategory.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (PartialOrder.toPreorder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteSemilatticeInf.toPartialOrder.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) (TopCat.topologicalSpace_coe.{u2} X)))))) (CategoryTheory.inducedFunctor.{u2, u1, u2} ι (TopologicalSpace.Opens.{u2} (CategoryTheory.Bundled.α.{u2, u2} TopologicalSpace.{u2} X) 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+Case conversion may be inaccurate. Consider using '#align Top.sheaf.extend_hom_app TopCat.Sheaf.extend_hom_appₓ'. -/
 @[simp]
 theorem extend_hom_app (α : (inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.1) (i : ι) :
     (restrictHomEquivHom F F' h α).app (op (B i)) = α.app (op i) :=
@@ -229,6 +339,12 @@ theorem extend_hom_app (α : (inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).
 
 include h
 
+/- warning: Top.sheaf.hom_ext -> TopCat.Sheaf.hom_ext is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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(CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))))))) C _inst_1 (CategoryTheory.Sheaf.val.{u1, u2, u1, u3} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (Opens.grothendieckTopology.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) C _inst_1 F')) (Opposite.op.{succ u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (B i)))) (CategoryTheory.NatTrans.app.{u1, u2, u1, u3} (Opposite.{succ u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))) (CategoryTheory.Category.opposite.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))))))) C _inst_1 F (CategoryTheory.Sheaf.val.{u1, u2, u1, u3} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (Opens.grothendieckTopology.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) C _inst_1 F') α (Opposite.op.{succ u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (B i))) (CategoryTheory.NatTrans.app.{u1, u2, u1, u3} (Opposite.{succ u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))) (CategoryTheory.Category.opposite.{u1, u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X))))))) C _inst_1 F (CategoryTheory.Sheaf.val.{u1, u2, u1, u3} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (Opens.grothendieckTopology.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) C _inst_1 F') β (Opposite.op.{succ u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (B i)))) -> (Eq.{max (succ u2) (succ u1)} (Quiver.Hom.{max (succ u2) (succ u1), max (max u3 u2) u1} (TopCat.Presheaf.{u1, u2, u3} C _inst_1 X) (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, max (max u3 u2) u1} (TopCat.Presheaf.{u1, u2, u3} C _inst_1 X) (CategoryTheory.Category.toCategoryStruct.{max u2 u1, max (max u3 u2) u1} (TopCat.Presheaf.{u1, u2, u3} C _inst_1 X) (TopCat.instCategoryPresheaf.{u1, u2, u3} C _inst_1 X))) F (CategoryTheory.Sheaf.val.{u1, u2, u1, u3} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (Preorder.smallCategory.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (PartialOrder.toPreorder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (TopologicalSpace.Opens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) (TopologicalSpace.Opens.instCompleteLatticeOpens.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)))))) (Opens.grothendieckTopology.{u1} (CategoryTheory.Bundled.α.{u1, u1} TopologicalSpace.{u1} X) (TopCat.topologicalSpace_coe.{u1} X)) C _inst_1 F')) α β))
+Case conversion may be inaccurate. Consider using '#align Top.sheaf.hom_ext TopCat.Sheaf.hom_extₓ'. -/
 theorem hom_ext {α β : F ⟶ F'.1} (he : ∀ i, α.app (op (B i)) = β.app (op (B i))) : α = β :=
   by
   apply (restrict_hom_equiv_hom F F' h).symm.Injective

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -58,17 +58,17 @@ variable (U : Opens X) (R : Presieve U)
 If `R` is a presieve in the grothendieck topology on `opens X`, the covering family associated to
 `R` really is _covering_, i.e. the union of all open sets equals `U`.
 -/
-theorem supᵢ_eq_of_mem_grothendieck (hR : Sieve.generate R ∈ Opens.grothendieckTopology X U) :
-    supᵢ (coveringOfPresieve U R) = U := by
+theorem iSup_eq_of_mem_grothendieck (hR : Sieve.generate R ∈ Opens.grothendieckTopology X U) :
+    iSup (coveringOfPresieve U R) = U := by
   apply le_antisymm
-  · refine' supᵢ_le _
+  · refine' iSup_le _
     intro f
     exact f.2.1.le
   intro x hxU
   rw [opens.mem_supr]
   obtain ⟨V, iVU, ⟨W, iVW, iWU, hiWU, -⟩, hxV⟩ := hR x hxU
   exact ⟨⟨W, ⟨iWU, hiWU⟩⟩, iVW.le hxV⟩
-#align Top.presheaf.covering_of_presieve.supr_eq_of_mem_grothendieck TopCat.Presheaf.coveringOfPresieve.supᵢ_eq_of_mem_grothendieck
+#align Top.presheaf.covering_of_presieve.supr_eq_of_mem_grothendieck TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieck
 
 end CoveringOfPresieve
 
@@ -81,8 +81,8 @@ def presieveOfCoveringAux {ι : Type v} (U : ι → Opens X) (Y : Opens X) : Pre
 #align Top.presheaf.presieve_of_covering_aux TopCat.Presheaf.presieveOfCoveringAux
 
 /-- Take `Y` to be `supr U` and obtain a presieve over `supr U`. -/
-def presieveOfCovering {ι : Type v} (U : ι → Opens X) : Presieve (supᵢ U) :=
-  presieveOfCoveringAux U (supᵢ U)
+def presieveOfCovering {ι : Type v} (U : ι → Opens X) : Presieve (iSup U) :=
+  presieveOfCoveringAux U (iSup U)
 #align Top.presheaf.presieve_of_covering TopCat.Presheaf.presieveOfCovering
 
 /-- Given a presieve `R` on `Y`, if we take its associated family of opens via
@@ -104,7 +104,7 @@ variable {ι : Type v} (U : ι → Opens X)
 /-- The sieve generated by `presieve_of_covering U` is a member of the grothendieck topology.
 -/
 theorem mem_grothendieckTopology :
-    Sieve.generate (presieveOfCovering U) ∈ Opens.grothendieckTopology X (supᵢ U) :=
+    Sieve.generate (presieveOfCovering U) ∈ Opens.grothendieckTopology X (iSup U) :=
   by
   intro x hx
   obtain ⟨i, hxi⟩ := opens.mem_supr.mp hx
@@ -114,18 +114,18 @@ theorem mem_grothendieckTopology :
 /-- An index `i : ι` can be turned into a dependent pair `(V, f)`, where `V` is an open set and
 `f : V ⟶ supr U` is a member of `presieve_of_covering U f`.
 -/
-def homOfIndex (i : ι) : ΣV, { f : V ⟶ supᵢ U // presieveOfCovering U f } :=
+def homOfIndex (i : ι) : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f } :=
   ⟨U i, Opens.leSupr U i, i, rfl⟩
 #align Top.presheaf.presieve_of_covering.hom_of_index TopCat.Presheaf.presieveOfCovering.homOfIndex
 
 /-- By using the axiom of choice, a dependent pair `(V, f)` where `f : V ⟶ supr U` is a member of
 `presieve_of_covering U f` can be turned into an index `i : ι`, such that `V = U i`.
 -/
-def indexOfHom (f : ΣV, { f : V ⟶ supᵢ U // presieveOfCovering U f }) : ι :=
+def indexOfHom (f : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f }) : ι :=
   f.2.2.some
 #align Top.presheaf.presieve_of_covering.index_of_hom TopCat.Presheaf.presieveOfCovering.indexOfHom
 
-theorem indexOfHom_spec (f : ΣV, { f : V ⟶ supᵢ U // presieveOfCovering U f }) :
+theorem indexOfHom_spec (f : ΣV, { f : V ⟶ iSup U // presieveOfCovering U f }) :
     f.1 = U (indexOfHom U f) :=
   f.2.2.choose_spec
 #align Top.presheaf.presieve_of_covering.index_of_hom_spec TopCat.Presheaf.presieveOfCovering.indexOfHom_spec

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

A PR analogous to #12338 and #12361: reformatting proofs following the multiple goals linter of #12339.

@@ -137,8 +137,9 @@ variable {X : TopCat} {ι : Type*}
 theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
     B.IsCoverDense (Opens.grothendieckTopology X) ↔ Opens.IsBasis (Set.range B.obj) := by
   rw [Opens.isBasis_iff_nbhd]
-  constructor; intro hd U x hx; rcases hd.1 U x hx with ⟨V, f, ⟨i, f₁, f₂, _⟩, hV⟩
-  exact ⟨B.obj i, ⟨i, rfl⟩, f₁.le hV, f₂.le⟩
+  constructor
+  · intro hd U x hx; rcases hd.1 U x hx with ⟨V, f, ⟨i, f₁, f₂, _⟩, hV⟩
+    exact ⟨B.obj i, ⟨i, rfl⟩, f₁.le hV, f₂.le⟩
   intro hb; constructor; intro U x hx; rcases hb hx with ⟨_, ⟨i, rfl⟩, hx, hi⟩
   exact ⟨B.obj i, ⟨⟨hi⟩⟩, ⟨⟨i, 𝟙 _, ⟨⟨hi⟩⟩, rfl⟩⟩, hx⟩
 #align Top.opens.cover_dense_iff_is_basis TopCat.Opens.coverDense_iff_isBasis

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -155,7 +155,6 @@ section OpenEmbedding
 open TopCat.Presheaf Opposite
 
 variable {C : Type u} [Category.{v} C]
-
 variable {X Y : TopCat.{w}} {f : X ⟶ Y} {F : Y.Presheaf C}
 
 theorem OpenEmbedding.compatiblePreserving (hf : OpenEmbedding f) :
@@ -228,9 +227,7 @@ namespace TopCat.Sheaf
 open TopCat Opposite
 
 variable {C : Type u} [Category.{v} C]
-
 variable {X : TopCat.{w}} {ι : Type*} {B : ι → Opens X}
-
 variable (F : X.Presheaf C) (F' : Sheaf C X) (h : Opens.IsBasis (Set.range B))
 
 /-- The empty component of a sheaf is terminal. -/

See Zulip thread for the discussion.

@@ -198,11 +198,11 @@ instance : RepresentablyFlat (Opens.map f) := by
   refine @IsCofiltered.mk _ _ ?_ ?_
   · constructor
     · intro V W
-      exact ⟨⟨⟨PUnit.unit⟩, V.right ⊓ W.right, homOfLE $ le_inf V.hom.le W.hom.le⟩,
+      exact ⟨⟨⟨PUnit.unit⟩, V.right ⊓ W.right, homOfLE <| le_inf V.hom.le W.hom.le⟩,
         StructuredArrow.homMk (homOfLE inf_le_left),
         StructuredArrow.homMk (homOfLE inf_le_right), trivial⟩
     · exact fun _ _ _ _ ↦ ⟨_, 𝟙 _, by simp [eq_iff_true_of_subsingleton]⟩
-  · exact ⟨StructuredArrow.mk $ show U ⟶ (Opens.map f).obj ⊤ from homOfLE le_top⟩
+  · exact ⟨StructuredArrow.mk <| show U ⟶ (Opens.map f).obj ⊤ from homOfLE le_top⟩
 
 theorem compatiblePreserving_opens_map :
     CompatiblePreserving (Opens.grothendieckTopology X) (Opens.map f) :=

This PR introduces the typeclass Functor.IsContinuous which says that the precomposition with a functor preserves the sheaf condition for Grothendieck topologies. It slightly refactors the previous main theorem about CoverPreserving and CompatiblePreserving functors: it now states that such functors are continuous. The pushforward functor for a continuous functor is defined under the Functor.IsContinuous assumption rather than the combination of both CoverPreserving and CompatiblePreserving. The property CoverLifting is renamed IsCocontinuous and it is made a class. The property IsCoverDense is also made a class.

Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>

@@ -18,8 +18,8 @@ topology, in preparation of connecting the sheaf condition on sites to the vario
 on spaces.
 
 We also specialize results about sheaves on sites to sheaves on spaces; we show that the inclusion
-functor from a topological basis to `TopologicalSpace.Opens` is `CoverDense`, that open maps
-induce `CoverPreserving` functors, and that open embeddings induce `CompatiblePreserving` functors.
+functor from a topological basis to `TopologicalSpace.Opens` is cover dense, that open maps
+induce cover preserving functors, and that open embeddings induce continuous functors.
 
 -/
 
@@ -135,7 +135,7 @@ namespace TopCat.Opens
 variable {X : TopCat} {ι : Type*}
 
 theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
-    CoverDense (Opens.grothendieckTopology X) B ↔ Opens.IsBasis (Set.range B.obj) := by
+    B.IsCoverDense (Opens.grothendieckTopology X) ↔ Opens.IsBasis (Set.range B.obj) := by
   rw [Opens.isBasis_iff_nbhd]
   constructor; intro hd U x hx; rcases hd.1 U x hx with ⟨V, f, ⟨i, f₁, f₂, _⟩, hV⟩
   exact ⟨B.obj i, ⟨i, rfl⟩, f₁.le hV, f₂.le⟩
@@ -144,7 +144,7 @@ theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
 #align Top.opens.cover_dense_iff_is_basis TopCat.Opens.coverDense_iff_isBasis
 
 theorem coverDense_inducedFunctor {B : ι → Opens X} (h : Opens.IsBasis (Set.range B)) :
-    CoverDense (Opens.grothendieckTopology X) (inducedFunctor B) :=
+    (inducedFunctor B).IsCoverDense (Opens.grothendieckTopology X)  :=
   (coverDense_iff_isBasis _).2 h
 #align Top.opens.cover_dense_induced_functor TopCat.Opens.coverDense_inducedFunctor
 
@@ -176,9 +176,18 @@ theorem IsOpenMap.coverPreserving (hf : IsOpenMap f) :
   exact ⟨_, hf.functor.map i, ⟨_, i, 𝟙 _, hV, rfl⟩, Set.mem_image_of_mem f hxV⟩
 #align is_open_map.cover_preserving IsOpenMap.coverPreserving
 
+
+lemma OpenEmbedding.functor_isContinuous (h : OpenEmbedding f) :
+    h.isOpenMap.functor.IsContinuous (Opens.grothendieckTopology X)
+      (Opens.grothendieckTopology Y) := by
+  apply Functor.isContinuous_of_coverPreserving
+  · exact h.compatiblePreserving
+  · exact h.isOpenMap.coverPreserving
+
 theorem TopCat.Presheaf.isSheaf_of_openEmbedding (h : OpenEmbedding f) (hF : F.IsSheaf) :
-    IsSheaf (h.isOpenMap.functor.op ⋙ F) :=
-  pullback_isSheaf_of_coverPreserving h.compatiblePreserving h.isOpenMap.coverPreserving ⟨_, hF⟩
+    IsSheaf (h.isOpenMap.functor.op ⋙ F) := by
+  have := h.functor_isContinuous
+  exact Functor.op_comp_isSheaf _ _ _ ⟨_, hF⟩
 #align Top.presheaf.is_sheaf_of_open_embedding TopCat.Presheaf.isSheaf_of_openEmbedding
 
 variable (f)
@@ -206,6 +215,12 @@ theorem coverPreserving_opens_map : CoverPreserving (Opens.grothendieckTopology
   obtain ⟨V, i, hi, hxV⟩ := hS (f x) hx
   exact ⟨_, (Opens.map f).map i, ⟨_, _, 𝟙 _, hi, Subsingleton.elim _ _⟩, hxV⟩
 
+instance : (Opens.map f).IsContinuous (Opens.grothendieckTopology Y)
+    (Opens.grothendieckTopology X) := by
+  apply Functor.isContinuous_of_coverPreserving
+  · exact compatiblePreserving_opens_map f
+  · exact coverPreserving_opens_map f
+
 end OpenEmbedding
 
 namespace TopCat.Sheaf
@@ -233,8 +248,10 @@ def isTerminalOfEqEmpty (F : X.Sheaf C) {U : Opens X} (h : U = ⊥) :
     is a sheaf on `X`, then a homomorphism between a presheaf `F` on `X` and `F'`
     is equivalent to a homomorphism between their restrictions to the indexing type
     `ι` of `B`, with the induced category structure on `ι`. -/
-def restrictHomEquivHom : ((inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.1) ≃ (F ⟶ F'.1) :=
-  @CoverDense.restrictHomEquivHom _ _ _ _ _ _ _ _ (Opens.coverDense_inducedFunctor h) _ F F'
+def restrictHomEquivHom :
+    ((inducedFunctor B).op ⋙ F ⟶ (inducedFunctor B).op ⋙ F'.1) ≃ (F ⟶ F'.1) :=
+  @Functor.IsCoverDense.restrictHomEquivHom _ _ _ _ _ _ _ _
+    (Opens.coverDense_inducedFunctor h) _ F F'
 #align Top.sheaf.restrict_hom_equiv_hom TopCat.Sheaf.restrictHomEquivHom
 
 @[simp]

Mathlib.Logic.Unique contains the line attribute [simp] eq_iff_true_of_subsingleton in ...:

https://github.com/leanprover-community/mathlib4/blob/96a11c7aac574c00370c2b3dab483cb676405c5d/Mathlib/Logic/Unique.lean#L255-L256

Despite what the in part may imply, this adds the lemma to the simp set "globally", including for downstream files; it is likely that attribute [local simp] eq_iff_true_of_subsingleton in ... was meant instead (or maybe scoped simp, but I think "scoped" refers to the current namespace). Indeed, the relevant lemma is not marked with @[simp] for possible slowness: https://github.com/leanprover/std4/blob/846e9e1d6bb534774d1acd2dc430e70987da3c18/Std/Logic.lean#L749. Adding it to the simp set causes the example at https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Regression.20in.20simp to slow down.

This PR changes this and fixes the relevant downstream simps. There was also one ocurrence of attribute [simp] FullSubcategory.comp_def FullSubcategory.id_def in in Mathlib.CategoryTheory.Monoidal.Subcategory but that was much easier to fix.

https://github.com/leanprover-community/mathlib4/blob/bc49eb9ba756a233370b4b68bcdedd60402f71ed/Mathlib/CategoryTheory/Monoidal/Subcategory.lean#L118-L119

@@ -192,7 +192,7 @@ instance : RepresentablyFlat (Opens.map f) := by
       exact ⟨⟨⟨PUnit.unit⟩, V.right ⊓ W.right, homOfLE $ le_inf V.hom.le W.hom.le⟩,
         StructuredArrow.homMk (homOfLE inf_le_left),
         StructuredArrow.homMk (homOfLE inf_le_right), trivial⟩
-    · exact fun _ _ _ _ ↦ ⟨_, 𝟙 _, by aesop⟩
+    · exact fun _ _ _ _ ↦ ⟨_, 𝟙 _, by simp [eq_iff_true_of_subsingleton]⟩
   · exact ⟨StructuredArrow.mk $ show U ⟶ (Opens.map f).obj ⊤ from homOfLE le_top⟩
 
 theorem compatiblePreserving_opens_map :

@@ -197,7 +197,7 @@ instance : RepresentablyFlat (Opens.map f) := by
 
 theorem compatiblePreserving_opens_map :
     CompatiblePreserving (Opens.grothendieckTopology X) (Opens.map f) :=
-compatiblePreservingOfFlat _ _
+  compatiblePreservingOfFlat _ _
 
 theorem coverPreserving_opens_map : CoverPreserving (Opens.grothendieckTopology Y)
     (Opens.grothendieckTopology X) (Opens.map f) := by

Port of https://github.com/leanprover-community/mathlib/pull/17561

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -181,6 +181,31 @@ theorem TopCat.Presheaf.isSheaf_of_openEmbedding (h : OpenEmbedding f) (hF : F.I
   pullback_isSheaf_of_coverPreserving h.compatiblePreserving h.isOpenMap.coverPreserving ⟨_, hF⟩
 #align Top.presheaf.is_sheaf_of_open_embedding TopCat.Presheaf.isSheaf_of_openEmbedding
 
+variable (f)
+
+instance : RepresentablyFlat (Opens.map f) := by
+  constructor
+  intro U
+  refine @IsCofiltered.mk _ _ ?_ ?_
+  · constructor
+    · intro V W
+      exact ⟨⟨⟨PUnit.unit⟩, V.right ⊓ W.right, homOfLE $ le_inf V.hom.le W.hom.le⟩,
+        StructuredArrow.homMk (homOfLE inf_le_left),
+        StructuredArrow.homMk (homOfLE inf_le_right), trivial⟩
+    · exact fun _ _ _ _ ↦ ⟨_, 𝟙 _, by aesop⟩
+  · exact ⟨StructuredArrow.mk $ show U ⟶ (Opens.map f).obj ⊤ from homOfLE le_top⟩
+
+theorem compatiblePreserving_opens_map :
+    CompatiblePreserving (Opens.grothendieckTopology X) (Opens.map f) :=
+compatiblePreservingOfFlat _ _
+
+theorem coverPreserving_opens_map : CoverPreserving (Opens.grothendieckTopology Y)
+    (Opens.grothendieckTopology X) (Opens.map f) := by
+  constructor
+  intro U S hS x hx
+  obtain ⟨V, i, hi, hxV⟩ := hS (f x) hx
+  exact ⟨_, (Opens.map f).map i, ⟨_, _, 𝟙 _, hi, Subsingleton.elim _ _⟩, hxV⟩
+
 end OpenEmbedding
 
 namespace TopCat.Sheaf

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -132,7 +132,7 @@ end TopCat.Presheaf
 
 namespace TopCat.Opens
 
-variable {X : TopCat} {ι : Type _}
+variable {X : TopCat} {ι : Type*}
 
 theorem coverDense_iff_isBasis [Category ι] (B : ι ⥤ Opens X) :
     CoverDense (Opens.grothendieckTopology X) B ↔ Opens.IsBasis (Set.range B.obj) := by
@@ -189,7 +189,7 @@ open TopCat Opposite
 
 variable {C : Type u} [Category.{v} C]
 
-variable {X : TopCat.{w}} {ι : Type _} {B : ι → Opens X}
+variable {X : TopCat.{w}} {ι : Type*} {B : ι → Opens X}
 
 variable (F : X.Presheaf C) (F' : Sheaf C X) (h : Opens.IsBasis (Set.range B))
 

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,16 +2,13 @@
 Copyright (c) 2021 Justus Springer. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Justus Springer
-
-! This file was ported from Lean 3 source module topology.sheaves.sheaf_condition.sites
-! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.CategoryTheory.Sites.Spaces
 import Mathlib.Topology.Sheaves.Sheaf
 import Mathlib.CategoryTheory.Sites.DenseSubsite
 
+#align_import topology.sheaves.sheaf_condition.sites from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
+
 /-!
 
 # Coverings and sieves; from sheaves on sites and sheaves on spaces