category_theory.sites.cover_preserving ⟷ Mathlib.CategoryTheory.Sites.CoverPreserving

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

@@ -182,7 +182,8 @@ theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type
 -/
 
 #print CategoryTheory.compatiblePreservingOfDownwardsClosed /-
-theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [Full F] [Faithful F]
+theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [CategoryTheory.Functor.Full F]
+    [CategoryTheory.Functor.Faithful F]
     (hF : ∀ {c : C} {d : D} (f : d ⟶ F.obj c), Σ c', F.obj c' ≅ d) : CompatiblePreserving K F :=
   by
   constructor

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

@@ -205,7 +205,7 @@ theorem pullback_isSheaf_of_coverPreserving {G : C ⥤ D} (hG₁ : CompatiblePre
     (hG₂ : CoverPreserving J K G) (ℱ : Sheaf K A) : Presheaf.IsSheaf J (G.op ⋙ ℱ.val) :=
   by
   intro X U S hS x hx
-  change family_of_elements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) _ at x 
+  change family_of_elements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) _ at x
   let H := ℱ.2 X _ (hG₂.cover_preserve hS)
   let hx' := hx.functor_pushforward hG₁ (sheaf_over ℱ X)
   constructor; swap

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

@@ -196,7 +196,6 @@ theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [Full F] [Faithful F
 #align category_theory.compatible_preserving_of_downwards_closed CategoryTheory.compatiblePreservingOfDownwardsClosed
 -/
 
-#print CategoryTheory.pullback_isSheaf_of_coverPreserving /-
 /-- If `G` is cover-preserving and compatible-preserving,
 then `G.op ⋙ _` pulls sheaves back to sheaves.
 
@@ -225,30 +224,28 @@ theorem pullback_isSheaf_of_coverPreserving {G : C ⥤ D} (hG₁ : CompatiblePre
     dsimp
     simp [hG₁.apply_map (sheaf_over ℱ X) hx h, ← hy f' h]
 #align category_theory.pullback_is_sheaf_of_cover_preserving CategoryTheory.pullback_isSheaf_of_coverPreserving
--/
 
-#print CategoryTheory.pullbackSheaf /-
 /-- The pullback of a sheaf along a cover-preserving and compatible-preserving functor. -/
 def pullbackSheaf {G : C ⥤ D} (hG₁ : CompatiblePreserving K G) (hG₂ : CoverPreserving J K G)
     (ℱ : Sheaf K A) : Sheaf J A :=
   ⟨G.op ⋙ ℱ.val, pullback_isSheaf_of_coverPreserving hG₁ hG₂ ℱ⟩
 #align category_theory.pullback_sheaf CategoryTheory.pullbackSheaf
--/
 
 variable (A)
 
-#print CategoryTheory.Sites.pullback /-
+#print CategoryTheory.Functor.sheafPushforwardContinuous /-
 /-- The induced functor from `Sheaf K A ⥤ Sheaf J A` given by `G.op ⋙ _`
 if `G` is cover-preserving and compatible-preserving.
 -/
 @[simps]
-def Sites.pullback {G : C ⥤ D} (hG₁ : CompatiblePreserving K G) (hG₂ : CoverPreserving J K G) :
-    Sheaf K A ⥤ Sheaf J A where
+def Functor.sheafPushforwardContinuous {G : C ⥤ D} (hG₁ : CompatiblePreserving K G)
+    (hG₂ : CoverPreserving J K G) : Sheaf K A ⥤ Sheaf J A
+    where
   obj ℱ := pullbackSheaf hG₁ hG₂ ℱ
   map _ _ f := ⟨((whiskeringLeft _ _ _).obj G.op).map f.val⟩
   map_id' ℱ := by ext1; apply ((whiskering_left _ _ _).obj G.op).map_id
   map_comp' _ _ _ f g := by ext1; apply ((whiskering_left _ _ _).obj G.op).map_comp
-#align category_theory.sites.pullback CategoryTheory.Sites.pullback
+#align category_theory.sites.pullback CategoryTheory.Functor.sheafPushforwardContinuous
 -/
 
 end CategoryTheory

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

@@ -3,9 +3,9 @@ Copyright (c) 2021 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
-import Mathbin.CategoryTheory.Functor.Flat
-import Mathbin.CategoryTheory.Sites.Sheaf
-import Mathbin.Tactic.ApplyFun
+import CategoryTheory.Functor.Flat
+import CategoryTheory.Sites.Sheaf
+import Tactic.ApplyFun
 
 #align_import category_theory.sites.cover_preserving from "leanprover-community/mathlib"@"781cb2eed038c4caf53bdbd8d20a95e5822d77df"
 

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

@@ -2,16 +2,13 @@
 Copyright (c) 2021 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module category_theory.sites.cover_preserving
-! leanprover-community/mathlib commit 781cb2eed038c4caf53bdbd8d20a95e5822d77df
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.CategoryTheory.Functor.Flat
 import Mathbin.CategoryTheory.Sites.Sheaf
 import Mathbin.Tactic.ApplyFun
 
+#align_import category_theory.sites.cover_preserving from "leanprover-community/mathlib"@"781cb2eed038c4caf53bdbd8d20a95e5822d77df"
+
 /-!
 # Cover-preserving functors between sites.
 

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

@@ -121,8 +121,7 @@ variable {J K} {G : C ⥤ D} (hG : CompatiblePreserving.{w} K G) (ℱ : SheafOfT
 
 variable {T : Presieve Z} {x : FamilyOfElements (G.op ⋙ ℱ.val) T} (h : x.Compatible)
 
-include h hG
-
+#print CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward /-
 /-- `compatible_preserving` functors indeed preserve compatible families. -/
 theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
     (x.functorPushforward G).Compatible :=
@@ -136,7 +135,9 @@ theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
   apply hG.compatible ℱ h _ _ hf₁ hf₂
   simpa using Eq
 #align category_theory.presieve.family_of_elements.compatible.functor_pushforward CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward
+-/
 
+#print CategoryTheory.CompatiblePreserving.apply_map /-
 @[simp]
 theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :
     x.functorPushforward G (G.map f) (image_mem_functorPushforward G T hf) = x f hf :=
@@ -146,11 +147,11 @@ theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :
     ⟨X, g, f', hg, eq⟩
   simpa using hG.compatible ℱ h f' (𝟙 _) hg hf (by simp [Eq])
 #align category_theory.compatible_preserving.apply_map CategoryTheory.CompatiblePreserving.apply_map
-
-omit h hG
+-/
 
 open Limits.WalkingCospan
 
+#print CategoryTheory.compatiblePreservingOfFlat /-
 theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
     (K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G :=
   by
@@ -181,7 +182,9 @@ theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type
   injection c'.π.naturality walking_cospan.hom.inr with _ e₂
   exact hx (c'.π.app left).right (c'.π.app right).right hg₁ hg₂ (e₁.symm.trans e₂)
 #align category_theory.compatible_preserving_of_flat CategoryTheory.compatiblePreservingOfFlat
+-/
 
+#print CategoryTheory.compatiblePreservingOfDownwardsClosed /-
 theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [Full F] [Faithful F]
     (hF : ∀ {c : C} {d : D} (f : d ⟶ F.obj c), Σ c', F.obj c' ≅ d) : CompatiblePreserving K F :=
   by
@@ -194,6 +197,7 @@ theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [Full F] [Faithful F
     hx (F.preimage <| e.hom ≫ f₁) (F.preimage <| e.hom ≫ f₂) hg₁ hg₂
       (F.map_injective <| by simpa using he)
 #align category_theory.compatible_preserving_of_downwards_closed CategoryTheory.compatiblePreservingOfDownwardsClosed
+-/
 
 #print CategoryTheory.pullback_isSheaf_of_coverPreserving /-
 /-- If `G` is cover-preserving and compatible-preserving,

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

@@ -183,7 +183,7 @@ theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type
 #align category_theory.compatible_preserving_of_flat CategoryTheory.compatiblePreservingOfFlat
 
 theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [Full F] [Faithful F]
-    (hF : ∀ {c : C} {d : D} (f : d ⟶ F.obj c), Σc', F.obj c' ≅ d) : CompatiblePreserving K F :=
+    (hF : ∀ {c : C} {d : D} (f : d ⟶ F.obj c), Σ c', F.obj c' ≅ d) : CompatiblePreserving K F :=
   by
   constructor
   introv hx he
@@ -205,7 +205,7 @@ theorem pullback_isSheaf_of_coverPreserving {G : C ⥤ D} (hG₁ : CompatiblePre
     (hG₂ : CoverPreserving J K G) (ℱ : Sheaf K A) : Presheaf.IsSheaf J (G.op ⋙ ℱ.val) :=
   by
   intro X U S hS x hx
-  change family_of_elements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) _ at x
+  change family_of_elements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) _ at x 
   let H := ℱ.2 X _ (hG₂.cover_preserve hS)
   let hx' := hx.functor_pushforward hG₁ (sheaf_over ℱ X)
   constructor; swap

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

@@ -123,12 +123,6 @@ variable {T : Presieve Z} {x : FamilyOfElements (G.op ⋙ ℱ.val) T} (h : x.Com
 
 include h hG
 
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-Case conversion may be inaccurate. Consider using '#align category_theory.presieve.family_of_elements.compatible.functor_pushforward CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforwardₓ'. -/
 /-- `compatible_preserving` functors indeed preserve compatible families. -/
 theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
     (x.functorPushforward G).Compatible :=
@@ -143,9 +137,6 @@ theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
   simpa using Eq
 #align category_theory.presieve.family_of_elements.compatible.functor_pushforward CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward
 
-/- warning: category_theory.compatible_preserving.apply_map -> CategoryTheory.CompatiblePreserving.apply_map is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.compatible_preserving.apply_map CategoryTheory.CompatiblePreserving.apply_mapₓ'. -/
 @[simp]
 theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :
     x.functorPushforward G (G.map f) (image_mem_functorPushforward G T hf) = x f hf :=
@@ -160,12 +151,6 @@ omit h hG
 
 open Limits.WalkingCospan
 
-/- warning: category_theory.compatible_preserving_of_flat -> CategoryTheory.compatiblePreservingOfFlat is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align category_theory.compatible_preserving_of_flat CategoryTheory.compatiblePreservingOfFlatₓ'. -/
 theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
     (K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G :=
   by
@@ -197,12 +182,6 @@ theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type
   exact hx (c'.π.app left).right (c'.π.app right).right hg₁ hg₂ (e₁.symm.trans e₂)
 #align category_theory.compatible_preserving_of_flat CategoryTheory.compatiblePreservingOfFlat
 
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-Case conversion may be inaccurate. Consider using '#align category_theory.compatible_preserving_of_downwards_closed CategoryTheory.compatiblePreservingOfDownwardsClosedₓ'. -/
 theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [Full F] [Faithful F]
     (hF : ∀ {c : C} {d : D} (f : d ⟶ F.obj c), Σc', F.obj c' ≅ d) : CompatiblePreserving K F :=
   by

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

@@ -180,16 +180,10 @@ theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type
     Then, it suffices to prove that it is compatible when restricted onto `u(c'.X.right)`.
     -/
   let c' := is_cofiltered.cone (structured_arrow_cone.to_diagram c ⋙ structured_arrow.pre _ _ _)
-  have eq₁ : f₁ = (c'.X.hom ≫ G.map (c'.π.app left).right) ≫ eq_to_hom (by simp) :=
-    by
-    erw [← (c'.π.app left).w]
-    dsimp
-    simp
-  have eq₂ : f₂ = (c'.X.hom ≫ G.map (c'.π.app right).right) ≫ eq_to_hom (by simp) :=
-    by
-    erw [← (c'.π.app right).w]
-    dsimp
-    simp
+  have eq₁ : f₁ = (c'.X.hom ≫ G.map (c'.π.app left).right) ≫ eq_to_hom (by simp) := by
+    erw [← (c'.π.app left).w]; dsimp; simp
+  have eq₂ : f₂ = (c'.X.hom ≫ G.map (c'.π.app right).right) ≫ eq_to_hom (by simp) := by
+    erw [← (c'.π.app right).w]; dsimp; simp
   conv_lhs => rw [eq₁]
   conv_rhs => rw [eq₂]
   simp only [op_comp, functor.map_comp, types_comp_apply, eq_to_hom_op, eq_to_hom_map]
@@ -272,12 +266,8 @@ def Sites.pullback {G : C ⥤ D} (hG₁ : CompatiblePreserving K G) (hG₂ : Cov
     Sheaf K A ⥤ Sheaf J A where
   obj ℱ := pullbackSheaf hG₁ hG₂ ℱ
   map _ _ f := ⟨((whiskeringLeft _ _ _).obj G.op).map f.val⟩
-  map_id' ℱ := by
-    ext1
-    apply ((whiskering_left _ _ _).obj G.op).map_id
-  map_comp' _ _ _ f g := by
-    ext1
-    apply ((whiskering_left _ _ _).obj G.op).map_comp
+  map_id' ℱ := by ext1; apply ((whiskering_left _ _ _).obj G.op).map_id
+  map_comp' _ _ _ f g := by ext1; apply ((whiskering_left _ _ _).obj G.op).map_comp
 #align category_theory.sites.pullback CategoryTheory.Sites.pullback
 -/
 

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

@@ -144,10 +144,7 @@ theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
 #align category_theory.presieve.family_of_elements.compatible.functor_pushforward CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward
 
 /- warning: category_theory.compatible_preserving.apply_map -> CategoryTheory.CompatiblePreserving.apply_map is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align category_theory.compatible_preserving.apply_map CategoryTheory.CompatiblePreserving.apply_mapₓ'. -/
 @[simp]
 theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 
 ! This file was ported from Lean 3 source module category_theory.sites.cover_preserving
-! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
+! leanprover-community/mathlib commit 781cb2eed038c4caf53bdbd8d20a95e5822d77df
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Tactic.ApplyFun
 /-!
 # Cover-preserving functors between sites.
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We define cover-preserving functors between sites as functors that push covering sieves to
 covering sieves. A cover-preserving and compatible-preserving functor `G : C ⥤ D` then pulls
 sheaves on `D` back to sheaves on `C` via `G.op ⋙ -`.

mathlib commit https://github.com/leanprover-community/mathlib/commit/28b2a92f2996d28e580450863c130955de0ed398

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 
 ! This file was ported from Lean 3 source module category_theory.sites.cover_preserving
-! leanprover-community/mathlib commit 781cb2eed038c4caf53bdbd8d20a95e5822d77df
+! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,9 +15,6 @@ import Mathbin.Tactic.ApplyFun
 /-!
 # Cover-preserving functors between sites.
 
-> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
-> Any changes to this file require a corresponding PR to mathlib4.
-
 We define cover-preserving functors between sites as functors that push covering sieves to
 covering sieves. A cover-preserving and compatible-preserving functor `G : C ⥤ D` then pulls
 sheaves on `D` back to sheaves on `C` via `G.op ⋙ -`.

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 
 ! This file was ported from Lean 3 source module category_theory.sites.cover_preserving
-! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
+! leanprover-community/mathlib commit 781cb2eed038c4caf53bdbd8d20a95e5822d77df
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Tactic.ApplyFun
 /-!
 # Cover-preserving functors between sites.
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We define cover-preserving functors between sites as functors that push covering sieves to
 covering sieves. A cover-preserving and compatible-preserving functor `G : C ⥤ D` then pulls
 sheaves on `D` back to sheaves on `C` via `G.op ⋙ -`.

mathlib commit https://github.com/leanprover-community/mathlib/commit/28b2a92f2996d28e580450863c130955de0ed398

@@ -68,6 +68,7 @@ variable (J : GrothendieckTopology C) (K : GrothendieckTopology D)
 
 variable {L : GrothendieckTopology A}
 
+#print CategoryTheory.CoverPreserving /-
 /-- A functor `G : (C, J) ⥤ (D, K)` between sites is *cover-preserving*
 if for all covering sieves `R` in `C`, `R.pushforward_functor G` is a covering sieve in `D`.
 -/
@@ -75,14 +76,18 @@ if for all covering sieves `R` in `C`, `R.pushforward_functor G` is a covering s
 structure CoverPreserving (G : C ⥤ D) : Prop where
   cover_preserve : ∀ {U : C} {S : Sieve U} (hS : S ∈ J U), S.functorPushforward G ∈ K (G.obj U)
 #align category_theory.cover_preserving CategoryTheory.CoverPreserving
+-/
 
+#print CategoryTheory.idCoverPreserving /-
 /-- The identity functor on a site is cover-preserving. -/
 theorem idCoverPreserving : CoverPreserving J J (𝟭 _) :=
   ⟨fun U S hS => by simpa using hS⟩
 #align category_theory.id_cover_preserving CategoryTheory.idCoverPreserving
+-/
 
 variable (J) (K)
 
+#print CategoryTheory.CoverPreserving.comp /-
 /-- The composition of two cover-preserving functors is cover-preserving. -/
 theorem CoverPreserving.comp {F} (hF : CoverPreserving J K F) {G} (hG : CoverPreserving K L G) :
     CoverPreserving J L (F ⋙ G) :=
@@ -90,7 +95,9 @@ theorem CoverPreserving.comp {F} (hF : CoverPreserving J K F) {G} (hG : CoverPre
     rw [sieve.functor_pushforward_comp]
     exact hG.cover_preserve (hF.cover_preserve hS)⟩
 #align category_theory.cover_preserving.comp CategoryTheory.CoverPreserving.comp
+-/
 
+#print CategoryTheory.CompatiblePreserving /-
 /-- A functor `G : (C, J) ⥤ (D, K)` between sites is called compatible preserving if for each
 compatible family of elements at `C` and valued in `G.op ⋙ ℱ`, and each commuting diagram
 `f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂`, `x g₁` and `x g₂` coincide when restricted via `fᵢ`.
@@ -105,6 +112,7 @@ structure CompatiblePreserving (K : GrothendieckTopology D) (G : C ⥤ D) : Prop
       {g₂ : Y₂ ⟶ Z} (hg₁ : T g₁) (hg₂ : T g₂) (eq : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂),
       ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)
 #align category_theory.compatible_preserving CategoryTheory.CompatiblePreserving
+-/
 
 variable {J K} {G : C ⥤ D} (hG : CompatiblePreserving.{w} K G) (ℱ : SheafOfTypes.{w} K) {Z : C}
 
@@ -112,6 +120,12 @@ variable {T : Presieve Z} {x : FamilyOfElements (G.op ⋙ ℱ.val) T} (h : x.Com
 
 include h hG
 
+/- warning: category_theory.presieve.family_of_elements.compatible.functor_pushforward -> CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward 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 category_theory.presieve.family_of_elements.compatible.functor_pushforward CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforwardₓ'. -/
 /-- `compatible_preserving` functors indeed preserve compatible families. -/
 theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
     (x.functorPushforward G).Compatible :=
@@ -126,6 +140,12 @@ theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
   simpa using Eq
 #align category_theory.presieve.family_of_elements.compatible.functor_pushforward CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward
 
+/- warning: category_theory.compatible_preserving.apply_map -> CategoryTheory.CompatiblePreserving.apply_map is a dubious translation:
+lean 3 declaration is
+  forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u2, u4} C] {D : Type.{u5}} [_inst_2 : CategoryTheory.Category.{u3, u5} D] {K : CategoryTheory.GrothendieckTopology.{u3, u5} D _inst_2} {G : CategoryTheory.Functor.{u2, u3, u4, u5} C _inst_1 D _inst_2}, (CategoryTheory.CompatiblePreserving.{u1, u2, u3, u4, u5} C _inst_1 D _inst_2 K G) -> (forall (ℱ : CategoryTheory.SheafOfTypes.{u1, u3, u5} D _inst_2 K) {Z : C} {T : CategoryTheory.Presieve.{u2, u4} C _inst_1 Z} {x : CategoryTheory.Presieve.FamilyOfElements.{u1, u2, u4} C _inst_1 Z (CategoryTheory.Functor.comp.{u2, u3, u1, u4, u5, succ u1} (Opposite.{succ u4} C) (CategoryTheory.Category.opposite.{u2, u4} C _inst_1) (Opposite.{succ u5} D) (CategoryTheory.Category.opposite.{u3, u5} D _inst_2) Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.Functor.op.{u2, u3, u4, u5} C _inst_1 D _inst_2 G) (CategoryTheory.SheafOfTypes.val.{u1, u3, u5} D _inst_2 K ℱ)) T}, (CategoryTheory.Presieve.FamilyOfElements.Compatible.{u1, u2, u4} C _inst_1 (CategoryTheory.Functor.comp.{u2, u3, u1, u4, u5, succ u1} (Opposite.{succ u4} C) (CategoryTheory.Category.opposite.{u2, u4} C _inst_1) (Opposite.{succ u5} D) (CategoryTheory.Category.opposite.{u3, u5} D _inst_2) Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.Functor.op.{u2, u3, u4, u5} C _inst_1 D _inst_2 G) (CategoryTheory.SheafOfTypes.val.{u1, u3, u5} D _inst_2 K ℱ)) Z T x) -> (forall {Y : C} {f : Quiver.Hom.{succ u2, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} C (CategoryTheory.Category.toCategoryStruct.{u2, u4} C _inst_1)) Y Z} (hf : T Y f), Eq.{succ u1} (CategoryTheory.Functor.obj.{u3, u1, u5, succ u1} (Opposite.{succ u5} D) (CategoryTheory.Category.opposite.{u3, u5} D _inst_2) Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.SheafOfTypes.val.{u1, u3, u5} D _inst_2 K ℱ) (Opposite.op.{succ u5} D (CategoryTheory.Functor.obj.{u2, u3, u4, u5} C _inst_1 D _inst_2 G Y))) (CategoryTheory.Presieve.FamilyOfElements.functorPushforward.{u1, u3, u2, u5, u4} D _inst_2 (CategoryTheory.SheafOfTypes.val.{u1, u3, u5} D _inst_2 K ℱ) C _inst_1 G Z T x (CategoryTheory.Functor.obj.{u2, u3, u4, u5} C _inst_1 D _inst_2 G Y) (CategoryTheory.Functor.map.{u2, u3, u4, u5} C _inst_1 D _inst_2 G Y Z f) (CategoryTheory.Presieve.image_mem_functorPushforward.{u2, u3, u4, u5} C _inst_1 D _inst_2 G Z Y T f hf)) (x Y f hf)))
+but is expected to have type
+  forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u2, u4} C] {D : Type.{u5}} [_inst_2 : CategoryTheory.Category.{u3, u5} D] {K : CategoryTheory.GrothendieckTopology.{u3, u5} D _inst_2} {G : CategoryTheory.Functor.{u2, u3, u4, u5} C _inst_1 D _inst_2}, (CategoryTheory.CompatiblePreserving.{u1, u2, u3, u4, u5} C _inst_1 D _inst_2 K G) -> (forall (ℱ : CategoryTheory.SheafOfTypes.{u1, u3, u5} D _inst_2 K) {Z : C} {T : CategoryTheory.Presieve.{u2, u4} C _inst_1 Z} {x : CategoryTheory.Presieve.FamilyOfElements.{u1, u2, u4} C _inst_1 Z (CategoryTheory.Functor.comp.{u2, u3, u1, u4, u5, succ u1} (Opposite.{succ u4} C) (CategoryTheory.Category.opposite.{u2, u4} C _inst_1) (Opposite.{succ u5} D) (CategoryTheory.Category.opposite.{u3, u5} D _inst_2) Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.Functor.op.{u2, u3, u4, u5} C _inst_1 D _inst_2 G) (CategoryTheory.SheafOfTypes.val.{u1, u3, u5} D _inst_2 K ℱ)) T}, (CategoryTheory.Presieve.FamilyOfElements.Compatible.{u1, u2, u4} C _inst_1 (CategoryTheory.Functor.comp.{u2, u3, u1, u4, u5, succ u1} (Opposite.{succ u4} C) (CategoryTheory.Category.opposite.{u2, u4} C _inst_1) (Opposite.{succ u5} D) (CategoryTheory.Category.opposite.{u3, u5} D _inst_2) Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.Functor.op.{u2, u3, u4, u5} C _inst_1 D _inst_2 G) (CategoryTheory.SheafOfTypes.val.{u1, u3, u5} D _inst_2 K ℱ)) Z T x) -> (forall {Y : C} {f : Quiver.Hom.{succ u2, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} C (CategoryTheory.Category.toCategoryStruct.{u2, u4} C _inst_1)) Y Z} (hf : T Y f), Eq.{succ u1} (Prefunctor.obj.{succ u3, succ u1, u5, succ u1} (Opposite.{succ u5} D) (CategoryTheory.CategoryStruct.toQuiver.{u3, u5} (Opposite.{succ u5} D) (CategoryTheory.Category.toCategoryStruct.{u3, u5} (Opposite.{succ u5} D) (CategoryTheory.Category.opposite.{u3, u5} D _inst_2))) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u3, u1, u5, succ u1} (Opposite.{succ u5} D) (CategoryTheory.Category.opposite.{u3, u5} D _inst_2) Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.SheafOfTypes.val.{u1, u3, u5} D _inst_2 K ℱ)) (Opposite.op.{succ u5} D (Prefunctor.obj.{succ u2, succ u3, u4, u5} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} C (CategoryTheory.Category.toCategoryStruct.{u2, u4} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u5} D (CategoryTheory.Category.toCategoryStruct.{u3, u5} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u2, u3, u4, u5} C _inst_1 D _inst_2 G) Y))) (CategoryTheory.Presieve.FamilyOfElements.functorPushforward.{u1, u3, u2, u5, u4} D _inst_2 (CategoryTheory.SheafOfTypes.val.{u1, u3, u5} D _inst_2 K ℱ) C _inst_1 G Z T x (Prefunctor.obj.{succ u2, succ u3, u4, u5} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} C (CategoryTheory.Category.toCategoryStruct.{u2, u4} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u5} D (CategoryTheory.Category.toCategoryStruct.{u3, u5} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u2, u3, u4, u5} C _inst_1 D _inst_2 G) Y) (Prefunctor.map.{succ u2, succ u3, u4, u5} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} C (CategoryTheory.Category.toCategoryStruct.{u2, u4} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u5} D (CategoryTheory.Category.toCategoryStruct.{u3, u5} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u2, u3, u4, u5} C _inst_1 D _inst_2 G) Y Z f) (CategoryTheory.Presieve.image_mem_functorPushforward.{u2, u3, u4, u5} C _inst_1 D _inst_2 G Z Y T f hf)) (x Y f hf)))
+Case conversion may be inaccurate. Consider using '#align category_theory.compatible_preserving.apply_map CategoryTheory.CompatiblePreserving.apply_mapₓ'. -/
 @[simp]
 theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :
     x.functorPushforward G (G.map f) (image_mem_functorPushforward G T hf) = x f hf :=
@@ -140,6 +160,12 @@ omit h hG
 
 open Limits.WalkingCospan
 
+/- warning: category_theory.compatible_preserving_of_flat -> CategoryTheory.compatiblePreservingOfFlat is a dubious translation:
+lean 3 declaration is
+  forall {C : Type.{u2}} [_inst_4 : CategoryTheory.Category.{u1, u2} C] {D : Type.{u2}} [_inst_5 : CategoryTheory.Category.{u1, u2} D] (K : CategoryTheory.GrothendieckTopology.{u1, u2} D _inst_5) (G : CategoryTheory.Functor.{u1, u1, u2, u2} C _inst_4 D _inst_5) [_inst_6 : CategoryTheory.RepresentablyFlat.{u1, u1, u2, u2} C _inst_4 D _inst_5 G], CategoryTheory.CompatiblePreserving.{u3, u1, u1, u2, u2} C _inst_4 D _inst_5 K G
+but is expected to have type
+  forall {C : Type.{u3}} [_inst_4 : CategoryTheory.Category.{u2, u3} C] {D : Type.{u3}} [_inst_5 : CategoryTheory.Category.{u2, u3} D] (K : CategoryTheory.GrothendieckTopology.{u2, u3} D _inst_5) (G : CategoryTheory.Functor.{u2, u2, u3, u3} C _inst_4 D _inst_5) [_inst_6 : CategoryTheory.RepresentablyFlat.{u2, u2, u3, u3} C _inst_4 D _inst_5 G], CategoryTheory.CompatiblePreserving.{u1, u2, u2, u3, u3} C _inst_4 D _inst_5 K G
+Case conversion may be inaccurate. Consider using '#align category_theory.compatible_preserving_of_flat CategoryTheory.compatiblePreservingOfFlatₓ'. -/
 theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
     (K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G :=
   by
@@ -177,6 +203,12 @@ theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type
   exact hx (c'.π.app left).right (c'.π.app right).right hg₁ hg₂ (e₁.symm.trans e₂)
 #align category_theory.compatible_preserving_of_flat CategoryTheory.compatiblePreservingOfFlat
 
+/- warning: category_theory.compatible_preserving_of_downwards_closed -> CategoryTheory.compatiblePreservingOfDownwardsClosed is a dubious translation:
+lean 3 declaration is
+  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] {K : CategoryTheory.GrothendieckTopology.{u2, u4} D _inst_2} (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) [_inst_4 : CategoryTheory.Full.{u1, u2, u3, u4} C _inst_1 D _inst_2 F] [_inst_5 : CategoryTheory.Faithful.{u1, u2, u3, u4} C _inst_1 D _inst_2 F], (forall {c : C} {d : D}, (Quiver.Hom.{succ u2, u4} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) d (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F c)) -> (Sigma.{u3, u2} C (fun (c' : C) => CategoryTheory.Iso.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F c') d))) -> (CategoryTheory.CompatiblePreserving.{u5, u1, u2, u3, u4} C _inst_1 D _inst_2 K F)
+but is expected to have type
+  forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u2, u4} C] {D : Type.{u5}} [_inst_2 : CategoryTheory.Category.{u3, u5} D] {K : CategoryTheory.GrothendieckTopology.{u3, u5} D _inst_2} (F : CategoryTheory.Functor.{u2, u3, u4, u5} C _inst_1 D _inst_2) [_inst_4 : CategoryTheory.Full.{u2, u3, u4, u5} C _inst_1 D _inst_2 F] [_inst_5 : CategoryTheory.Faithful.{u2, u3, u4, u5} C _inst_1 D _inst_2 F], (forall {c : C} {d : D}, (Quiver.Hom.{succ u3, u5} D (CategoryTheory.CategoryStruct.toQuiver.{u3, u5} D (CategoryTheory.Category.toCategoryStruct.{u3, u5} D _inst_2)) d (Prefunctor.obj.{succ u2, succ u3, u4, u5} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} C (CategoryTheory.Category.toCategoryStruct.{u2, u4} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u5} D (CategoryTheory.Category.toCategoryStruct.{u3, u5} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u2, u3, u4, u5} C _inst_1 D _inst_2 F) c)) -> (Sigma.{u4, u3} C (fun (c' : C) => CategoryTheory.Iso.{u3, u5} D _inst_2 (Prefunctor.obj.{succ u2, succ u3, u4, u5} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} C (CategoryTheory.Category.toCategoryStruct.{u2, u4} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u5} D (CategoryTheory.Category.toCategoryStruct.{u3, u5} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u2, u3, u4, u5} C _inst_1 D _inst_2 F) c') d))) -> (CategoryTheory.CompatiblePreserving.{u1, u2, u3, u4, u5} C _inst_1 D _inst_2 K F)
+Case conversion may be inaccurate. Consider using '#align category_theory.compatible_preserving_of_downwards_closed CategoryTheory.compatiblePreservingOfDownwardsClosedₓ'. -/
 theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [Full F] [Faithful F]
     (hF : ∀ {c : C} {d : D} (f : d ⟶ F.obj c), Σc', F.obj c' ≅ d) : CompatiblePreserving K F :=
   by
@@ -190,6 +222,7 @@ theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [Full F] [Faithful F
       (F.map_injective <| by simpa using he)
 #align category_theory.compatible_preserving_of_downwards_closed CategoryTheory.compatiblePreservingOfDownwardsClosed
 
+#print CategoryTheory.pullback_isSheaf_of_coverPreserving /-
 /-- If `G` is cover-preserving and compatible-preserving,
 then `G.op ⋙ _` pulls sheaves back to sheaves.
 
@@ -218,15 +251,19 @@ theorem pullback_isSheaf_of_coverPreserving {G : C ⥤ D} (hG₁ : CompatiblePre
     dsimp
     simp [hG₁.apply_map (sheaf_over ℱ X) hx h, ← hy f' h]
 #align category_theory.pullback_is_sheaf_of_cover_preserving CategoryTheory.pullback_isSheaf_of_coverPreserving
+-/
 
+#print CategoryTheory.pullbackSheaf /-
 /-- The pullback of a sheaf along a cover-preserving and compatible-preserving functor. -/
 def pullbackSheaf {G : C ⥤ D} (hG₁ : CompatiblePreserving K G) (hG₂ : CoverPreserving J K G)
     (ℱ : Sheaf K A) : Sheaf J A :=
   ⟨G.op ⋙ ℱ.val, pullback_isSheaf_of_coverPreserving hG₁ hG₂ ℱ⟩
 #align category_theory.pullback_sheaf CategoryTheory.pullbackSheaf
+-/
 
 variable (A)
 
+#print CategoryTheory.Sites.pullback /-
 /-- The induced functor from `Sheaf K A ⥤ Sheaf J A` given by `G.op ⋙ _`
 if `G` is cover-preserving and compatible-preserving.
 -/
@@ -242,6 +279,7 @@ def Sites.pullback {G : C ⥤ D} (hG₁ : CompatiblePreserving K G) (hG₂ : Cov
     ext1
     apply ((whiskering_left _ _ _).obj G.op).map_comp
 #align category_theory.sites.pullback CategoryTheory.Sites.pullback
+-/
 
 end CategoryTheory
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/738054fa93d43512da144ec45ce799d18fd44248

@@ -4,13 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 
 ! This file was ported from Lean 3 source module category_theory.sites.cover_preserving
-! leanprover-community/mathlib commit e2e38c005fc6f715502490da6cb0ec84df9ed228
+! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathbin.CategoryTheory.Sites.Limits
 import Mathbin.CategoryTheory.Functor.Flat
-import Mathbin.CategoryTheory.Limits.Preserves.Filtered
+import Mathbin.CategoryTheory.Sites.Sheaf
+import Mathbin.Tactic.ApplyFun
 
 /-!
 # Cover-preserving functors between sites.

mathlib commit https://github.com/leanprover-community/mathlib/commit/21e3562c5e12d846c7def5eff8cdbc520d7d4936

@@ -4,14 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 
 ! This file was ported from Lean 3 source module category_theory.sites.cover_preserving
-! leanprover-community/mathlib commit 85d6221d32c37e68f05b2e42cde6cee658dae5e9
+! leanprover-community/mathlib commit e2e38c005fc6f715502490da6cb0ec84df9ed228
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.CategoryTheory.Sites.Limits
 import Mathbin.CategoryTheory.Functor.Flat
 import Mathbin.CategoryTheory.Limits.Preserves.Filtered
-import Mathbin.CategoryTheory.Sites.LeftExact
 
 /-!
 # Cover-preserving functors between sites.
@@ -30,10 +29,6 @@ if it pushes compatible families of elements to compatible families.
 compatible-preserving functor.
 * `category_theory.sites.pullback`: the induced functor `Sheaf K A ⥤ Sheaf J A` for a
 cover-preserving and compatible-preserving functor `G : (C, J) ⥤ (D, K)`.
-* `category_theory.sites.pushforward`: the induced functor `Sheaf J A ⥤ Sheaf K A` for a
-cover-preserving and compatible-preserving functor `G : (C, J) ⥤ (D, K)`.
-* `category_theory.sites.pushforward`: the induced functor `Sheaf J A ⥤ Sheaf K A` for a
-cover-preserving and compatible-preserving functor `G : (C, J) ⥤ (D, K)`.
 
 ## Main results
 
@@ -250,53 +245,3 @@ def Sites.pullback {G : C ⥤ D} (hG₁ : CompatiblePreserving K G) (hG₂ : Cov
 
 end CategoryTheory
 
-namespace CategoryTheory
-
-variable {C : Type v₁} [SmallCategory C] {D : Type v₁} [SmallCategory D]
-
-variable (A : Type u₂) [Category.{v₁} A]
-
-variable (J : GrothendieckTopology C) (K : GrothendieckTopology D)
-
-instance [HasLimits A] : CreatesLimits (sheafToPresheaf J A) :=
-  CategoryTheory.Sheaf.CategoryTheory.SheafToPresheaf.CategoryTheory.createsLimits.{u₂, v₁, v₁}
-
--- The assumptions so that we have sheafification
-variable [ConcreteCategory.{v₁} A] [PreservesLimits (forget A)] [HasColimits A] [HasLimits A]
-
-variable [PreservesFilteredColimits (forget A)] [ReflectsIsomorphisms (forget A)]
-
-attribute [local instance] reflects_limits_of_reflects_isomorphisms
-
-instance {X : C} : IsCofiltered (J.cover X) :=
-  inferInstance
-
-/-- The pushforward functor `Sheaf J A ⥤ Sheaf K A` associated to a functor `G : C ⥤ D` in the
-same direction as `G`. -/
-@[simps]
-def Sites.pushforward (G : C ⥤ D) : Sheaf J A ⥤ Sheaf K A :=
-  sheafToPresheaf J A ⋙ lan G.op ⋙ presheafToSheaf K A
-#align category_theory.sites.pushforward CategoryTheory.Sites.pushforward
-
-instance (G : C ⥤ D) [RepresentablyFlat G] : PreservesFiniteLimits (Sites.pushforward A J K G) :=
-  by
-  apply (config := { instances := false }) comp_preserves_finite_limits
-  · infer_instance
-  apply (config := { instances := false }) comp_preserves_finite_limits
-  · apply CategoryTheory.lanPreservesFiniteLimitsOfFlat
-  · apply CategoryTheory.presheafToSheaf.Limits.preservesFiniteLimits.{u₂, v₁, v₁}
-    infer_instance
-
-/-- The pushforward functor is left adjoint to the pullback functor. -/
-def Sites.pullbackPushforwardAdjunction {G : C ⥤ D} (hG₁ : CompatiblePreserving K G)
-    (hG₂ : CoverPreserving J K G) : Sites.pushforward A J K G ⊣ Sites.pullback A hG₁ hG₂ :=
-  ((Lan.adjunction A G.op).comp (sheafificationAdjunction K A)).restrictFullyFaithful
-    (sheafToPresheaf J A) (𝟭 _)
-    (NatIso.ofComponents (fun _ => Iso.refl _) fun _ _ _ =>
-      (Category.comp_id _).trans (Category.id_comp _).symm)
-    (NatIso.ofComponents (fun _ => Iso.refl _) fun _ _ _ =>
-      (Category.comp_id _).trans (Category.id_comp _).symm)
-#align category_theory.sites.pullback_pushforward_adjunction CategoryTheory.Sites.pullbackPushforwardAdjunction
-
-end CategoryTheory
-

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -290,7 +290,7 @@ instance (G : C ⥤ D) [RepresentablyFlat G] : PreservesFiniteLimits (Sites.push
 /-- The pushforward functor is left adjoint to the pullback functor. -/
 def Sites.pullbackPushforwardAdjunction {G : C ⥤ D} (hG₁ : CompatiblePreserving K G)
     (hG₂ : CoverPreserving J K G) : Sites.pushforward A J K G ⊣ Sites.pullback A hG₁ hG₂ :=
-  ((lan.adjunction A G.op).comp (sheafificationAdjunction K A)).restrictFullyFaithful
+  ((Lan.adjunction A G.op).comp (sheafificationAdjunction K A)).restrictFullyFaithful
     (sheafToPresheaf J A) (𝟭 _)
     (NatIso.ofComponents (fun _ => Iso.refl _) fun _ _ _ =>
       (Category.comp_id _).trans (Category.id_comp _).symm)

mathlib commit https://github.com/leanprover-community/mathlib/commit/3b267e70a936eebb21ab546f49a8df34dd300b25

@@ -82,9 +82,9 @@ structure CoverPreserving (G : C ⥤ D) : Prop where
 #align category_theory.cover_preserving CategoryTheory.CoverPreserving
 
 /-- The identity functor on a site is cover-preserving. -/
-theorem id_coverPreserving : CoverPreserving J J (𝟭 _) :=
+theorem idCoverPreserving : CoverPreserving J J (𝟭 _) :=
   ⟨fun U S hS => by simpa using hS⟩
-#align category_theory.id_cover_preserving CategoryTheory.id_coverPreserving
+#align category_theory.id_cover_preserving CategoryTheory.idCoverPreserving
 
 variable (J) (K)
 

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

Reference the newly created issues #12094 and #12096, as well as the pre-existing #5171. Change all references to #10927 to #5171. Some of these changes were not labelled as "porting note"; change this for good measure.

@@ -63,7 +63,7 @@ variable {L : GrothendieckTopology A}
 /-- A functor `G : (C, J) ⥤ (D, K)` between sites is *cover-preserving*
 if for all covering sieves `R` in `C`, `R.functorPushforward G` is a covering sieve in `D`.
 -/
--- Porting note: removed `@[nolint has_nonempty_instance]`
+-- Porting note(#5171): removed `@[nolint has_nonempty_instance]`
 structure CoverPreserving (G : C ⥤ D) : Prop where
   cover_preserve : ∀ {U : C} {S : Sieve U} (_ : S ∈ J U), S.functorPushforward G ∈ K (G.obj U)
 #align category_theory.cover_preserving CategoryTheory.CoverPreserving
@@ -87,7 +87,7 @@ compatible family of elements at `C` and valued in `G.op ⋙ ℱ`, and each comm
 This is actually stronger than merely preserving compatible families because of the definition of
 `functorPushforward` used.
 -/
--- Porting note: this doesn't work yet @[nolint has_nonempty_instance]
+-- Porting note(#5171): linter not ported yet @[nolint has_nonempty_instance]
 structure CompatiblePreserving (K : GrothendieckTopology D) (G : C ⥤ D) : Prop where
   compatible :
     ∀ (ℱ : SheafOfTypes.{w} K) {Z} {T : Presieve Z} {x : FamilyOfElements (G.op ⋙ ℱ.val) T}

These notions on functors are now Functor.Full, Functor.Faithful, Functor.EssSurj, Functor.IsEquivalence, Functor.ReflectsIsomorphisms. Deprecated aliases are introduced for the previous names.

@@ -158,7 +158,7 @@ theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type
   exact hx (c'.π.app left).right (c'.π.app right).right hg₁ hg₂ (e₁.symm.trans e₂)
 #align category_theory.compatible_preserving_of_flat CategoryTheory.compatiblePreservingOfFlat
 
-theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [Full F] [Faithful F]
+theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [F.Full] [F.Faithful]
     (hF : ∀ {c : C} {d : D} (_ : d ⟶ F.obj c), Σc', F.obj c' ≅ d) : CompatiblePreserving K F := by
   constructor
   introv hx he

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)

@@ -56,11 +56,8 @@ open CategoryTheory Opposite CategoryTheory.Presieve.FamilyOfElements CategoryTh
 namespace CategoryTheory
 
 variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D] (F : C ⥤ D)
-
 variable {A : Type u₃} [Category.{v₃} A]
-
 variable (J : GrothendieckTopology C) (K : GrothendieckTopology D)
-
 variable {L : GrothendieckTopology A}
 
 /-- A functor `G : (C, J) ⥤ (D, K)` between sites is *cover-preserving*
@@ -100,7 +97,6 @@ structure CompatiblePreserving (K : GrothendieckTopology D) (G : C ⥤ D) : Prop
 #align category_theory.compatible_preserving CategoryTheory.CompatiblePreserving
 
 variable {J K} {G : C ⥤ D} (hG : CompatiblePreserving.{w} K G) (ℱ : SheafOfTypes.{w} K) {Z : C}
-
 variable {T : Presieve Z} {x : FamilyOfElements (G.op ⋙ ℱ.val) T} (h : x.Compatible)
 
 /-- `CompatiblePreserving` functors indeed preserve compatible families. -/

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -66,7 +66,7 @@ variable {L : GrothendieckTopology A}
 /-- A functor `G : (C, J) ⥤ (D, K)` between sites is *cover-preserving*
 if for all covering sieves `R` in `C`, `R.functorPushforward G` is a covering sieve in `D`.
 -/
--- porting note: removed `@[nolint has_nonempty_instance]`
+-- Porting note: removed `@[nolint has_nonempty_instance]`
 structure CoverPreserving (G : C ⥤ D) : Prop where
   cover_preserve : ∀ {U : C} {S : Sieve U} (_ : S ∈ J U), S.functorPushforward G ∈ K (G.obj U)
 #align category_theory.cover_preserving CategoryTheory.CoverPreserving
@@ -90,7 +90,7 @@ compatible family of elements at `C` and valued in `G.op ⋙ ℱ`, and each comm
 This is actually stronger than merely preserving compatible families because of the definition of
 `functorPushforward` used.
 -/
--- porting note: this doesn't work yet @[nolint has_nonempty_instance]
+-- Porting note: this doesn't work yet @[nolint has_nonempty_instance]
 structure CompatiblePreserving (K : GrothendieckTopology D) (G : C ⥤ D) : Prop where
   compatible :
     ∀ (ℱ : SheafOfTypes.{w} K) {Z} {T : Presieve Z} {x : FamilyOfElements (G.op ⋙ ℱ.val) T}
@@ -151,7 +151,7 @@ theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type
   conv_lhs => rw [eq₁]
   conv_rhs => rw [eq₂]
   simp only [op_comp, Functor.map_comp, types_comp_apply, eqToHom_op, eqToHom_map]
-  apply congr_arg -- porting note: was `congr 1` which for some reason doesn't do anything here
+  apply congr_arg -- Porting note: was `congr 1` which for some reason doesn't do anything here
   -- despite goal being of the form f a = f b, with f=`ℱ.val.map (Quiver.Hom.op c'.pt.hom)`
   /-
     Since everything now falls in the image of `u`,

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -142,11 +142,11 @@ theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type
   let c' := IsCofiltered.cone (c.toStructuredArrow ⋙ StructuredArrow.pre _ _ _)
   have eq₁ : f₁ = (c'.pt.hom ≫ G.map (c'.π.app left).right) ≫ eqToHom (by simp) := by
     erw [← (c'.π.app left).w]
-    dsimp
+    dsimp [c]
     simp
   have eq₂ : f₂ = (c'.pt.hom ≫ G.map (c'.π.app right).right) ≫ eqToHom (by simp) := by
     erw [← (c'.π.app right).w]
-    dsimp
+    dsimp [c]
     simp
   conv_lhs => rw [eq₁]
   conv_rhs => rw [eq₂]

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Joachim Breitner <mail@joachim-breitner.de>

@@ -120,7 +120,7 @@ theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
 theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :
     x.functorPushforward G (G.map f) (image_mem_functorPushforward G T hf) = x f hf := by
   unfold FamilyOfElements.functorPushforward
-  rcases e₁ : getFunctorPushforwardStructure (image_mem_functorPushforward G T hf) with
+  rcases getFunctorPushforwardStructure (image_mem_functorPushforward G T hf) with
     ⟨X, g, f', hg, eq⟩
   simpa using hG.compatible ℱ h f' (𝟙 _) hg hf (by simp [eq])
 #align category_theory.compatible_preserving.apply_map CategoryTheory.CompatiblePreserving.apply_map

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>

@@ -9,33 +9,39 @@ import Mathlib.Tactic.ApplyFun
 
 #align_import category_theory.sites.cover_preserving from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
 /-!
-# Cover-preserving functors between sites.
+# Cover-preserving and continuous functors between sites.
 
-We define cover-preserving functors between sites as functors that push covering sieves to
-covering sieves. A cover-preserving and compatible-preserving functor `G : C ⥤ D` then pulls
-sheaves on `D` back to sheaves on `C` via `G.op ⋙ -`.
+We define the notion of continuous functor between sites: these are functors `G` such that
+the precomposition with `G.op` preserves sheaves of types (and actually sheaves in any
+category).
+
+In order to show that a functor is continuous, we define cover-preserving functors
+between sites as functors that push covering sieves to covering sieves.
+Then, a cover-preserving and compatible-preserving functor is continuous.
 
 ## Main definitions
 
+* `CategoryTheory.Functor.IsContinuous`: a functor between sites is continuous if the
+precomposition with this functor preserves sheaves.
 * `CategoryTheory.CoverPreserving`: a functor between sites is cover-preserving if it
 pushes covering sieves to covering sieves
 * `CategoryTheory.CompatiblePreserving`: a functor between sites is compatible-preserving
 if it pushes compatible families of elements to compatible families.
-* `CategoryTheory.pullbackSheaf`: the pullback of a sheaf along a cover-preserving and
-compatible-preserving functor.
-* `CategoryTheory.Sites.pullback`: the induced functor `Sheaf K A ⥤ Sheaf J A` for a
-cover-preserving and compatible-preserving functor `G : (C, J) ⥤ (D, K)`.
+* `CategoryTheory.Functor.sheafPushforwardContinuous`: the induced functor
+`Sheaf K A ⥤ Sheaf J A` for a continuous functor `G : (C, J) ⥤ (D, K)`. In case this is
+part of a morphism of sites, this would be understood as the pushforward functor
+even though it goes in the opposite direction as the functor `G`.
 
 ## Main results
 
-- `CategoryTheory.pullback_isSheaf_of_coverPreserving`: If `G : C ⥤ D` is
-cover-preserving and compatible-preserving, then `G ⋙ -` (`uᵖ`) as a functor
-`(Dᵒᵖ ⥤ A) ⥤ (Cᵒᵖ ⥤ A)` of presheaves maps sheaves to sheaves.
+- `CategoryTheory.isContinuous_of_coverPreserving`: If `G : C ⥤ D` is
+cover-preserving and compatible-preserving, then `G` is a continuous functor,
+i.e. `G.op ⋙ -` as a functor `(Dᵒᵖ ⥤ A) ⥤ (Cᵒᵖ ⥤ A)` of presheaves maps sheaves to sheaves.
 
 ## References
 
 * [Elephant]: *Sketches of an Elephant*, P. T. Johnstone: C2.3.
-* https://stacks.math.columbia.edu/tag/00WW
+* https://stacks.math.columbia.edu/tag/00WU
 
 -/
 
@@ -49,7 +55,7 @@ open CategoryTheory Opposite CategoryTheory.Presieve.FamilyOfElements CategoryTh
 
 namespace CategoryTheory
 
-variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
+variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D] (F : C ⥤ D)
 
 variable {A : Type u₃} [Category.{v₃} A]
 
@@ -86,7 +92,7 @@ This is actually stronger than merely preserving compatible families because of
 -/
 -- porting note: this doesn't work yet @[nolint has_nonempty_instance]
 structure CompatiblePreserving (K : GrothendieckTopology D) (G : C ⥤ D) : Prop where
-  Compatible :
+  compatible :
     ∀ (ℱ : SheafOfTypes.{w} K) {Z} {T : Presieve Z} {x : FamilyOfElements (G.op ⋙ ℱ.val) T}
       (_ : x.Compatible) {Y₁ Y₂} {X} (f₁ : X ⟶ G.obj Y₁) (f₂ : X ⟶ G.obj Y₂) {g₁ : Y₁ ⟶ Z}
       {g₂ : Y₂ ⟶ Z} (hg₁ : T g₁) (hg₂ : T g₂) (_ : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂),
@@ -106,7 +112,7 @@ theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
   rcases getFunctorPushforwardStructure H₂ with ⟨X₂, f₂, h₂, hf₂, rfl⟩
   suffices ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂) by
     simpa using this
-  apply hG.Compatible ℱ h _ _ hf₁ hf₂
+  apply hG.compatible ℱ h _ _ hf₁ hf₂
   simpa using eq
 #align category_theory.presieve.family_of_elements.compatible.functor_pushforward CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward
 
@@ -116,7 +122,7 @@ theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :
   unfold FamilyOfElements.functorPushforward
   rcases e₁ : getFunctorPushforwardStructure (image_mem_functorPushforward G T hf) with
     ⟨X, g, f', hg, eq⟩
-  simpa using hG.Compatible ℱ h f' (𝟙 _) hg hf (by simp [eq])
+  simpa using hG.compatible ℱ h f' (𝟙 _) hg hf (by simp [eq])
 #align category_theory.compatible_preserving.apply_map CategoryTheory.CompatiblePreserving.apply_map
 
 open Limits.WalkingCospan
@@ -168,57 +174,85 @@ theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [Full F] [Faithful F
       (F.map_injective <| by simpa using he)
 #align category_theory.compatible_preserving_of_downwards_closed CategoryTheory.compatiblePreservingOfDownwardsClosed
 
-/-- If `G` is cover-preserving and compatible-preserving,
-then `G.op ⋙ _` pulls sheaves back to sheaves.
+variable (J K)
 
-This result is basically <https://stacks.math.columbia.edu/tag/00WW>.
--/
-theorem pullback_isSheaf_of_coverPreserving {G : C ⥤ D} (hG₁ : CompatiblePreserving.{v₃} K G)
-    (hG₂ : CoverPreserving J K G) (ℱ : Sheaf K A) : Presheaf.IsSheaf J (G.op ⋙ ℱ.val) := by
-  intro X U S hS x hx
-  change FamilyOfElements (G.op ⋙ ℱ.val ⋙ coyoneda.obj (op X)) _ at x
-  let H := ℱ.2 X _ (hG₂.cover_preserve hS)
-  let hx' := hx.functorPushforward hG₁ (sheafOver ℱ X)
-  constructor; swap
-  · apply H.amalgamate (x.functorPushforward G)
-    exact hx'
-  constructor
-  · intro V f hf
-    convert H.isAmalgamation hx' (G.map f) (image_mem_functorPushforward G S hf)
-    rw [hG₁.apply_map (sheafOver ℱ X) hx]
-  · intro y hy
-    refine'
-      H.isSeparatedFor _ y _ _ (H.isAmalgamation (hx.functorPushforward hG₁ (sheafOver ℱ X)))
-    rintro V f ⟨Z, f', g', h, rfl⟩
-    -- porting note: didn't need coercion (S : Presieve U) in Lean 3
-    erw [FamilyOfElements.comp_of_compatible (S.functorPushforward G) hx'
-        (image_mem_functorPushforward G (S : Presieve U) h) g']
-    dsimp
-    simp [hG₁.apply_map (sheafOver ℱ X) hx h, ← hy f' h]
-#align category_theory.pullback_is_sheaf_of_cover_preserving CategoryTheory.pullback_isSheaf_of_coverPreserving
+/-- A functor `F` is continuous if the precomposition with `F.op` sends sheaves of `Type w`
+to sheaves. -/
+class Functor.IsContinuous : Prop where
+  op_comp_isSheafOfTypes (G : SheafOfTypes.{w} K) : Presieve.IsSheaf J (F.op ⋙ G.val)
+
+lemma Functor.op_comp_isSheafOfTypes [Functor.IsContinuous.{w} F J K]
+    (G : SheafOfTypes.{w} K) :
+    Presieve.IsSheaf J (F.op ⋙ G.val) :=
+  Functor.IsContinuous.op_comp_isSheafOfTypes _
+
+lemma Functor.isContinuous_of_iso {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂)
+    (J : GrothendieckTopology C) (K : GrothendieckTopology D)
+    [Functor.IsContinuous.{w} F₁ J K] : Functor.IsContinuous.{w} F₂ J K where
+  op_comp_isSheafOfTypes G :=
+    Presieve.isSheaf_iso J (isoWhiskerRight (NatIso.op e.symm) _)
+      (F₁.op_comp_isSheafOfTypes J K G)
 
-/-- The pullback of a sheaf along a cover-preserving and compatible-preserving functor. -/
-def pullbackSheaf {G : C ⥤ D} (hG₁ : CompatiblePreserving K G) (hG₂ : CoverPreserving J K G)
-    (ℱ : Sheaf K A) : Sheaf J A :=
-  ⟨G.op ⋙ ℱ.val, pullback_isSheaf_of_coverPreserving hG₁ hG₂ ℱ⟩
-#align category_theory.pullback_sheaf CategoryTheory.pullbackSheaf
+instance Functor.isContinuous_id : Functor.IsContinuous.{w} (𝟭 C) J J where
+  op_comp_isSheafOfTypes G := G.2
 
-variable (A)
+lemma Functor.isContinuous_comp (F₁ : C ⥤ D) (F₂ : D ⥤ A) (J : GrothendieckTopology C)
+    (K : GrothendieckTopology D) (L : GrothendieckTopology A)
+    [Functor.IsContinuous.{w} F₁ J K] [Functor.IsContinuous.{w} F₂ K L] :
+    Functor.IsContinuous.{w} (F₁ ⋙ F₂) J L where
+  op_comp_isSheafOfTypes G := F₁.op_comp_isSheafOfTypes J K ⟨_, F₂.op_comp_isSheafOfTypes K L G⟩
 
-/-- The induced functor from `Sheaf K A ⥤ Sheaf J A` given by `G.op ⋙ _`
-if `G` is cover-preserving and compatible-preserving.
+lemma Functor.isContinuous_comp' {F₁ : C ⥤ D} {F₂ : D ⥤ A} {F₁₂ : C ⥤ A}
+    (e : F₁ ⋙ F₂ ≅ F₁₂) (J : GrothendieckTopology C)
+    (K : GrothendieckTopology D) (L : GrothendieckTopology A)
+    [Functor.IsContinuous.{w} F₁ J K] [Functor.IsContinuous.{w} F₂ K L] :
+    Functor.IsContinuous.{w} F₁₂ J L := by
+  have := Functor.isContinuous_comp F₁ F₂ J K L
+  apply Functor.isContinuous_of_iso e
+
+lemma Functor.op_comp_isSheaf [Functor.IsContinuous.{v₃} F J K] (G : Sheaf K A) :
+    Presheaf.IsSheaf J (F.op ⋙ G.val) :=
+  fun T => F.op_comp_isSheafOfTypes J K ⟨_, G.cond T⟩
+
+variable {F J K}
+
+/-- If `F` is cover-preserving and compatible-preserving,
+then `F` is a continuous functor.
+
+This result is basically <https://stacks.math.columbia.edu/tag/00WW>.
+-/
+lemma Functor.isContinuous_of_coverPreserving (hF₁ : CompatiblePreserving.{w} K F)
+    (hF₂ : CoverPreserving J K F) : Functor.IsContinuous.{w} F J K where
+  op_comp_isSheafOfTypes G X S hS x hx := by
+    apply exists_unique_of_exists_of_unique
+    · have H := G.2 _ (hF₂.cover_preserve hS)
+      exact ⟨H.amalgamate (x.functorPushforward F) (hx.functorPushforward hF₁),
+        fun V f hf => (H.isAmalgamation (hx.functorPushforward hF₁) (F.map f) _).trans
+          (hF₁.apply_map _ hx hf)⟩
+    · intro y₁ y₂ hy₁ hy₂
+      apply (Presieve.isSeparated_of_isSheaf _ _ G.cond _ (hF₂.cover_preserve hS)).ext
+      rintro Y _ ⟨Z, g, h, hg, rfl⟩
+      dsimp
+      simp only [Functor.map_comp, types_comp_apply]
+      have H := (hy₁ g hg).trans (hy₂ g hg).symm
+      dsimp at H
+      rw [H]
+
+variable (F J K A)
+
+/-- The induced functor `Sheaf K A ⥤ Sheaf J A` given by `G.op ⋙ _`
+if `G` is a continuous functor.
 -/
-@[simps]
-def Sites.pullback {G : C ⥤ D} (hG₁ : CompatiblePreserving K G) (hG₂ : CoverPreserving J K G) :
+def Functor.sheafPushforwardContinuous [Functor.IsContinuous.{v₃} F J K] :
     Sheaf K A ⥤ Sheaf J A where
-  obj ℱ := pullbackSheaf hG₁ hG₂ ℱ
-  map f := ⟨((whiskeringLeft _ _ _).obj G.op).map f.val⟩
+  obj ℱ := ⟨F.op ⋙ ℱ.val, F.op_comp_isSheaf J K ℱ⟩
+  map f := ⟨((whiskeringLeft _ _ _).obj F.op).map f.val⟩
   map_id ℱ := by
     ext1
-    apply ((whiskeringLeft _ _ _).obj G.op).map_id
+    apply ((whiskeringLeft _ _ _).obj F.op).map_id
   map_comp f g := by
     ext1
-    apply ((whiskeringLeft _ _ _).obj G.op).map_comp
-#align category_theory.sites.pullback CategoryTheory.Sites.pullback
+    apply ((whiskeringLeft _ _ _).obj F.op).map_comp
+#align category_theory.sites.pullback CategoryTheory.Functor.sheafPushforwardContinuous
 
 end CategoryTheory

After making this PR I noticed that this already existed in Functor/Flat.lean, so I unified the two developments.

@@ -133,7 +133,7 @@ theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type
     over it since `StructuredArrow W u` is cofiltered.
     Then, it suffices to prove that it is compatible when restricted onto `u(c'.X.right)`.
     -/
-  let c' := IsCofiltered.cone (StructuredArrowCone.toDiagram c ⋙ StructuredArrow.pre _ _ _)
+  let c' := IsCofiltered.cone (c.toStructuredArrow ⋙ StructuredArrow.pre _ _ _)
   have eq₁ : f₁ = (c'.pt.hom ≫ G.map (c'.π.app left).right) ≫ eqToHom (by simp) := by
     erw [← (c'.π.app left).w]
     dsimp

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2021 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module category_theory.sites.cover_preserving
-! leanprover-community/mathlib commit f0c8bf9245297a541f468be517f1bde6195105e9
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.CategoryTheory.Functor.Flat
 import Mathlib.CategoryTheory.Sites.Sheaf
 import Mathlib.Tactic.ApplyFun
+
+#align_import category_theory.sites.cover_preserving from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
 /-!
 # Cover-preserving functors between sites.
 

@@ -26,7 +26,7 @@ pushes covering sieves to covering sieves
 if it pushes compatible families of elements to compatible families.
 * `CategoryTheory.pullbackSheaf`: the pullback of a sheaf along a cover-preserving and
 compatible-preserving functor.
-* `category_theory.Sites.pullback`: the induced functor `Sheaf K A ⥤ Sheaf J A` for a
+* `CategoryTheory.Sites.pullback`: the induced functor `Sheaf K A ⥤ Sheaf J A` for a
 cover-preserving and compatible-preserving functor `G : (C, J) ⥤ (D, K)`.
 
 ## Main results
@@ -47,15 +47,8 @@ universe w v₁ v₂ v₃ u₁ u₂ u₃
 
 noncomputable section
 
-open CategoryTheory
-
-open Opposite
-
-open CategoryTheory.Presieve.FamilyOfElements
-
-open CategoryTheory.Presieve
-
-open CategoryTheory.Limits
+open CategoryTheory Opposite CategoryTheory.Presieve.FamilyOfElements CategoryTheory.Presieve
+  CategoryTheory.Limits
 
 namespace CategoryTheory
 
@@ -68,7 +61,7 @@ variable (J : GrothendieckTopology C) (K : GrothendieckTopology D)
 variable {L : GrothendieckTopology A}
 
 /-- A functor `G : (C, J) ⥤ (D, K)` between sites is *cover-preserving*
-if for all covering sieves `R` in `C`, `R.pushforward_functor G` is a covering sieve in `D`.
+if for all covering sieves `R` in `C`, `R.functorPushforward G` is a covering sieve in `D`.
 -/
 -- porting note: removed `@[nolint has_nonempty_instance]`
 structure CoverPreserving (G : C ⥤ D) : Prop where
@@ -80,9 +73,6 @@ theorem idCoverPreserving : CoverPreserving J J (𝟭 _) :=
   ⟨fun hS => by simpa using hS⟩
 #align category_theory.id_cover_preserving CategoryTheory.idCoverPreserving
 
--- porting note: this line is not needed as the variables are already explicit
--- variable (J) (K)
-
 /-- The composition of two cover-preserving functors is cover-preserving. -/
 theorem CoverPreserving.comp {F} (hF : CoverPreserving J K F) {G} (hG : CoverPreserving K L G) :
     CoverPreserving J L (F ⋙ G) :=
@@ -95,7 +85,7 @@ theorem CoverPreserving.comp {F} (hF : CoverPreserving J K F) {G} (hG : CoverPre
 compatible family of elements at `C` and valued in `G.op ⋙ ℱ`, and each commuting diagram
 `f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂`, `x g₁` and `x g₂` coincide when restricted via `fᵢ`.
 This is actually stronger than merely preserving compatible families because of the definition of
-`functor_pushforward` used.
+`functorPushforward` used.
 -/
 -- porting note: this doesn't work yet @[nolint has_nonempty_instance]
 structure CompatiblePreserving (K : GrothendieckTopology D) (G : C ⥤ D) : Prop where
@@ -110,18 +100,15 @@ variable {J K} {G : C ⥤ D} (hG : CompatiblePreserving.{w} K G) (ℱ : SheafOfT
 
 variable {T : Presieve Z} {x : FamilyOfElements (G.op ⋙ ℱ.val) T} (h : x.Compatible)
 
--- porting note: commenting out `include`
--- include h hG
-
-/-- `compatible_preserving` functors indeed preserve compatible families. -/
+/-- `CompatiblePreserving` functors indeed preserve compatible families. -/
 theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
     (x.functorPushforward G).Compatible := by
   rintro Z₁ Z₂ W g₁ g₂ f₁' f₂' H₁ H₂ eq
   unfold FamilyOfElements.functorPushforward
   rcases getFunctorPushforwardStructure H₁ with ⟨X₁, f₁, h₁, hf₁, rfl⟩
   rcases getFunctorPushforwardStructure H₂ with ⟨X₂, f₂, h₂, hf₂, rfl⟩
-  suffices : ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂)
-  simpa using this
+  suffices ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂) by
+    simpa using this
   apply hG.Compatible ℱ h _ _ hf₁ hf₂
   simpa using eq
 #align category_theory.presieve.family_of_elements.compatible.functor_pushforward CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward
@@ -135,9 +122,6 @@ theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :
   simpa using hG.Compatible ℱ h f' (𝟙 _) hg hf (by simp [eq])
 #align category_theory.compatible_preserving.apply_map CategoryTheory.CompatiblePreserving.apply_map
 
--- porting note: commenting out `omit`
--- omit h hG
-
 open Limits.WalkingCospan
 
 theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
@@ -149,7 +133,7 @@ theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type
     (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
   /-
     This can then be viewed as a cospan of structured arrows, and we may obtain an arbitrary cone
-    over it since `structured_arrow W u` is cofiltered.
+    over it since `StructuredArrow W u` is cofiltered.
     Then, it suffices to prove that it is compatible when restricted onto `u(c'.X.right)`.
     -/
   let c' := IsCofiltered.cone (StructuredArrowCone.toDiagram c ⋙ StructuredArrow.pre _ _ _)

Straightforward.