category_theory.sites.dense_subsite ⟷ Mathlib.CategoryTheory.Sites.DenseSubsite

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

@@ -532,14 +532,14 @@ theorem CategoryTheory.Functor.IsCoverDense.compatiblePreserving
 #align category_theory.cover_dense.compatible_preserving CategoryTheory.Functor.IsCoverDense.compatiblePreserving
 -/
 
-#print CategoryTheory.Functor.IsCoverDense.fullSheafPushforwardContinuous /-
-noncomputable instance CategoryTheory.Functor.IsCoverDense.fullSheafPushforwardContinuous
+#print CategoryTheory.Functor.IsCoverDense.full_sheafPushforwardContinuous /-
+noncomputable instance CategoryTheory.Functor.IsCoverDense.full_sheafPushforwardContinuous
     [CategoryTheory.Functor.Faithful G] (Hp : CoverPreserving J K G) :
     CategoryTheory.Functor.Full (Functor.sheafPushforwardContinuous A H.CompatiblePreserving Hp)
     where
   preimage ℱ ℱ' α := ⟨H.sheafHom α.val⟩
   witness' ℱ ℱ' α := Sheaf.Hom.ext _ _ <| H.sheafHom_restrict_eq α.val
-#align category_theory.cover_dense.sites.pullback.full CategoryTheory.Functor.IsCoverDense.fullSheafPushforwardContinuous
+#align category_theory.cover_dense.sites.pullback.full CategoryTheory.Functor.IsCoverDense.full_sheafPushforwardContinuous
 -/
 
 #print CategoryTheory.Functor.IsCoverDense.faithful_sheafPushforwardContinuous /-

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

@@ -128,9 +128,9 @@ theorem CategoryTheory.Functor.IsCoverDense.ext (H : CategoryTheory.Functor.IsCo
 -/
 
 #print CategoryTheory.Functor.IsCoverDense.functorPullback_pushforward_covering /-
-theorem CategoryTheory.Functor.IsCoverDense.functorPullback_pushforward_covering [Full G]
-    (H : CategoryTheory.Functor.IsCoverDense K G) {X : C} (T : K (G.obj X)) :
-    (T.val.functorPullback G).functorPushforward G ∈ K (G.obj X) :=
+theorem CategoryTheory.Functor.IsCoverDense.functorPullback_pushforward_covering
+    [CategoryTheory.Functor.Full G] (H : CategoryTheory.Functor.IsCoverDense K G) {X : C}
+    (T : K (G.obj X)) : (T.val.functorPullback G).functorPushforward G ∈ K (G.obj X) :=
   by
   refine' K.superset_covering _ (K.bind_covering T.property fun Y f Hf => H.is_cover Y)
   rintro Y _ ⟨Z, _, f, hf, ⟨W, g, f', ⟨rfl⟩⟩, rfl⟩
@@ -173,7 +173,7 @@ theorem CategoryTheory.Functor.IsCoverDense.sheaf_eq_amalgamation (ℱ : Sheaf K
 #align category_theory.cover_dense.sheaf_eq_amalgamation CategoryTheory.Functor.IsCoverDense.sheaf_eq_amalgamation
 -/
 
-variable [Full G]
+variable [CategoryTheory.Functor.Full G]
 
 namespace Types
 
@@ -516,8 +516,9 @@ theorem CategoryTheory.Functor.IsCoverDense.iso_of_restrict_iso {ℱ ℱ' : Shea
 
 #print CategoryTheory.Functor.IsCoverDense.compatiblePreserving /-
 /-- A fully faithful cover-dense functor preserves compatible families. -/
-theorem CategoryTheory.Functor.IsCoverDense.compatiblePreserving [Faithful G] :
-    CompatiblePreserving K G := by
+theorem CategoryTheory.Functor.IsCoverDense.compatiblePreserving
+    [CategoryTheory.Functor.Faithful G] : CompatiblePreserving K G :=
+  by
   constructor
   intro ℱ Z T x hx Y₁ Y₂ X f₁ f₂ g₁ g₂ hg₁ hg₂ eq
   apply H.ext
@@ -533,8 +534,8 @@ theorem CategoryTheory.Functor.IsCoverDense.compatiblePreserving [Faithful G] :
 
 #print CategoryTheory.Functor.IsCoverDense.fullSheafPushforwardContinuous /-
 noncomputable instance CategoryTheory.Functor.IsCoverDense.fullSheafPushforwardContinuous
-    [Faithful G] (Hp : CoverPreserving J K G) :
-    Full (Functor.sheafPushforwardContinuous A H.CompatiblePreserving Hp)
+    [CategoryTheory.Functor.Faithful G] (Hp : CoverPreserving J K G) :
+    CategoryTheory.Functor.Full (Functor.sheafPushforwardContinuous A H.CompatiblePreserving Hp)
     where
   preimage ℱ ℱ' α := ⟨H.sheafHom α.val⟩
   witness' ℱ ℱ' α := Sheaf.Hom.ext _ _ <| H.sheafHom_restrict_eq α.val
@@ -542,9 +543,9 @@ noncomputable instance CategoryTheory.Functor.IsCoverDense.fullSheafPushforwardC
 -/
 
 #print CategoryTheory.Functor.IsCoverDense.faithful_sheafPushforwardContinuous /-
-instance CategoryTheory.Functor.IsCoverDense.faithful_sheafPushforwardContinuous [Faithful G]
-    (Hp : CoverPreserving J K G) :
-    Faithful (Functor.sheafPushforwardContinuous A H.CompatiblePreserving Hp)
+instance CategoryTheory.Functor.IsCoverDense.faithful_sheafPushforwardContinuous
+    [CategoryTheory.Functor.Faithful G] (Hp : CoverPreserving J K G) :
+    CategoryTheory.Functor.Faithful (Functor.sheafPushforwardContinuous A H.CompatiblePreserving Hp)
     where map_injective' := by
     intro ℱ ℱ' α β e
     ext1
@@ -564,7 +565,7 @@ open CategoryTheory
 
 variable {C D : Type u} [Category.{v} C] [Category.{v} D]
 
-variable {G : C ⥤ D} [Full G] [Faithful G]
+variable {G : C ⥤ D} [CategoryTheory.Functor.Full G] [CategoryTheory.Functor.Faithful G]
 
 variable {J : GrothendieckTopology C} {K : GrothendieckTopology D}
 

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

@@ -352,7 +352,7 @@ noncomputable def CategoryTheory.Functor.IsCoverDense.sheafCoyonedaHom
     apply sheaf_eq_amalgamation
     apply H.is_cover
     intro Y' f' hf'
-    change unop X ⟶ ℱ.obj (op (unop _)) at x 
+    change unop X ⟶ ℱ.obj (op (unop _)) at x
     dsimp
     simp only [pushforward_family, functor.comp_map, coyoneda_obj_map, hom_over_app, category.assoc]
     congr 1
@@ -548,8 +548,8 @@ instance CategoryTheory.Functor.IsCoverDense.faithful_sheafPushforwardContinuous
     where map_injective' := by
     intro ℱ ℱ' α β e
     ext1
-    apply_fun fun e => e.val at e 
-    dsimp at e 
+    apply_fun fun e => e.val at e
+    dsimp at e
     rw [← H.sheaf_hom_eq α.val, ← H.sheaf_hom_eq β.val, e]
 #align category_theory.cover_dense.sites.pullback.faithful CategoryTheory.Functor.IsCoverDense.faithful_sheafPushforwardContinuous
 -/

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

@@ -96,15 +96,15 @@ theorem Presieve.in_coverByImage (G : C ⥤ D) {X : D} {Y : C} (f : G.obj Y ⟶
 #align category_theory.presieve.in_cover_by_image CategoryTheory.Presieve.in_coverByImage
 -/
 
-#print CategoryTheory.CoverDense /-
+#print CategoryTheory.Functor.IsCoverDense /-
 /-- A functor `G : (C, J) ⥤ (D, K)` is called `cover_dense` if for each object in `D`,
   there exists a covering sieve in `D` that factors through images of `G`.
 
 This definition can be found in https://ncatlab.org/nlab/show/dense+sub-site Definition 2.2.
 -/
-structure CoverDense (K : GrothendieckTopology D) (G : C ⥤ D) : Prop where
+structure CategoryTheory.Functor.IsCoverDense (K : GrothendieckTopology D) (G : C ⥤ D) : Prop where
   is_cover : ∀ U : D, Sieve.coverByImage G U ∈ K U
-#align category_theory.cover_dense CategoryTheory.CoverDense
+#align category_theory.cover_dense CategoryTheory.Functor.IsCoverDense
 -/
 
 open Presieve Opposite
@@ -113,22 +113,24 @@ namespace CoverDense
 
 variable {K}
 
-variable {A : Type _} [Category A] {G : C ⥤ D} (H : CoverDense K G)
+variable {A : Type _} [Category A] {G : C ⥤ D} (H : CategoryTheory.Functor.IsCoverDense K G)
 
-#print CategoryTheory.CoverDense.ext /-
+#print CategoryTheory.Functor.IsCoverDense.ext /-
 -- this is not marked with `@[ext]` because `H` can not be inferred from the type
-theorem ext (H : CoverDense K G) (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.obj (op X)}
+theorem CategoryTheory.Functor.IsCoverDense.ext (H : CategoryTheory.Functor.IsCoverDense K G)
+    (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.obj (op X)}
     (h : ∀ ⦃Y : C⦄ (f : G.obj Y ⟶ X), ℱ.val.map f.op s = ℱ.val.map f.op t) : s = t :=
   by
   apply (ℱ.cond (sieve.cover_by_image G X) (H.is_cover X)).IsSeparatedFor.ext
   rintro Y _ ⟨Z, f₁, f₂, ⟨rfl⟩⟩
   simp [h f₂]
-#align category_theory.cover_dense.ext CategoryTheory.CoverDense.ext
+#align category_theory.cover_dense.ext CategoryTheory.Functor.IsCoverDense.ext
 -/
 
-#print CategoryTheory.CoverDense.functorPullback_pushforward_covering /-
-theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X : C}
-    (T : K (G.obj X)) : (T.val.functorPullback G).functorPushforward G ∈ K (G.obj X) :=
+#print CategoryTheory.Functor.IsCoverDense.functorPullback_pushforward_covering /-
+theorem CategoryTheory.Functor.IsCoverDense.functorPullback_pushforward_covering [Full G]
+    (H : CategoryTheory.Functor.IsCoverDense K G) {X : C} (T : K (G.obj X)) :
+    (T.val.functorPullback G).functorPushforward G ∈ K (G.obj X) :=
   by
   refine' K.superset_covering _ (K.bind_covering T.property fun Y f Hf => H.is_cover Y)
   rintro Y _ ⟨Z, _, f, hf, ⟨W, g, f', ⟨rfl⟩⟩, rfl⟩
@@ -136,37 +138,39 @@ theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X :
   constructor
   · simpa using T.val.downward_closed hf f'
   · simp
-#align category_theory.cover_dense.functor_pullback_pushforward_covering CategoryTheory.CoverDense.functorPullback_pushforward_covering
+#align category_theory.cover_dense.functor_pullback_pushforward_covering CategoryTheory.Functor.IsCoverDense.functorPullback_pushforward_covering
 -/
 
-#print CategoryTheory.CoverDense.homOver /-
+#print CategoryTheory.Functor.IsCoverDense.homOver /-
 /-- (Implementation). Given an hom between the pullbacks of two sheaves, we can whisker it with
 `coyoneda` to obtain an hom between the pullbacks of the sheaves of maps from `X`.
 -/
 @[simps]
-def homOver {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A} (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) (X : A) :
+def CategoryTheory.Functor.IsCoverDense.homOver {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A}
+    (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) (X : A) :
     G.op ⋙ ℱ ⋙ coyoneda.obj (op X) ⟶ G.op ⋙ (sheafOver ℱ' X).val :=
   whiskerRight α (coyoneda.obj (op X))
-#align category_theory.cover_dense.hom_over CategoryTheory.CoverDense.homOver
+#align category_theory.cover_dense.hom_over CategoryTheory.Functor.IsCoverDense.homOver
 -/
 
-#print CategoryTheory.CoverDense.isoOver /-
+#print CategoryTheory.Functor.IsCoverDense.isoOver /-
 /-- (Implementation). Given an iso between the pullbacks of two sheaves, we can whisker it with
 `coyoneda` to obtain an iso between the pullbacks of the sheaves of maps from `X`.
 -/
 @[simps]
-def isoOver {ℱ ℱ' : Sheaf K A} (α : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) (X : A) :
+def CategoryTheory.Functor.IsCoverDense.isoOver {ℱ ℱ' : Sheaf K A}
+    (α : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) (X : A) :
     G.op ⋙ (sheafOver ℱ X).val ≅ G.op ⋙ (sheafOver ℱ' X).val :=
   isoWhiskerRight α (coyoneda.obj (op X))
-#align category_theory.cover_dense.iso_over CategoryTheory.CoverDense.isoOver
+#align category_theory.cover_dense.iso_over CategoryTheory.Functor.IsCoverDense.isoOver
 -/
 
-#print CategoryTheory.CoverDense.sheaf_eq_amalgamation /-
-theorem sheaf_eq_amalgamation (ℱ : Sheaf K A) {X : A} {U : D} {T : Sieve U} (hT)
-    (x : FamilyOfElements _ T) (hx) (t) (h : x.IsAmalgamation t) :
+#print CategoryTheory.Functor.IsCoverDense.sheaf_eq_amalgamation /-
+theorem CategoryTheory.Functor.IsCoverDense.sheaf_eq_amalgamation (ℱ : Sheaf K A) {X : A} {U : D}
+    {T : Sieve U} (hT) (x : FamilyOfElements _ T) (hx) (t) (h : x.IsAmalgamation t) :
     t = (ℱ.cond X T hT).amalgamate x hx :=
   (ℱ.cond X T hT).IsSeparatedFor x t _ h ((ℱ.cond X T hT).IsAmalgamation hx)
-#align category_theory.cover_dense.sheaf_eq_amalgamation CategoryTheory.CoverDense.sheaf_eq_amalgamation
+#align category_theory.cover_dense.sheaf_eq_amalgamation CategoryTheory.Functor.IsCoverDense.sheaf_eq_amalgamation
 -/
 
 variable [Full G]
@@ -175,20 +179,22 @@ namespace Types
 
 variable {ℱ : Dᵒᵖ ⥤ Type v} {ℱ' : SheafOfTypes.{v} K} (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val)
 
-#print CategoryTheory.CoverDense.Types.pushforwardFamily /-
+#print CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily /-
 /--
 (Implementation). Given a section of `ℱ` on `X`, we can obtain a family of elements valued in `ℱ'`
 that is defined on a cover generated by the images of `G`. -/
 @[simp, nolint unused_arguments]
-noncomputable def pushforwardFamily {X} (x : ℱ.obj (op X)) :
-    FamilyOfElements ℱ'.val (coverByImage G X) := fun Y f hf =>
+noncomputable def CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily {X}
+    (x : ℱ.obj (op X)) : FamilyOfElements ℱ'.val (coverByImage G X) := fun Y f hf =>
   ℱ'.val.map hf.some.lift.op <| α.app (op _) (ℱ.map hf.some.map.op x : _)
-#align category_theory.cover_dense.types.pushforward_family CategoryTheory.CoverDense.Types.pushforwardFamily
+#align category_theory.cover_dense.types.pushforward_family CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily
 -/
 
-#print CategoryTheory.CoverDense.Types.pushforwardFamily_compatible /-
+#print CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_compatible /-
 /-- (Implementation). The `pushforward_family` defined is compatible. -/
-theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) : (pushforwardFamily α x).Compatible :=
+theorem CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_compatible {X}
+    (x : ℱ.obj (op X)) :
+    (CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily α x).Compatible :=
   by
   intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ e
   apply H.ext
@@ -204,20 +210,26 @@ theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) : (pushforwardFami
     ℱ.map_comp, G.image_preimage]
   congr 3
   simp [e]
-#align category_theory.cover_dense.types.pushforward_family_compatible CategoryTheory.CoverDense.Types.pushforwardFamily_compatible
+#align category_theory.cover_dense.types.pushforward_family_compatible CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_compatible
 -/
 
-#print CategoryTheory.CoverDense.Types.appHom /-
+#print CategoryTheory.Functor.IsCoverDense.Types.appHom /-
 /-- (Implementation). The morphism `ℱ(X) ⟶ ℱ'(X)` given by gluing the `pushforward_family`. -/
-noncomputable def appHom (X : D) : ℱ.obj (op X) ⟶ ℱ'.val.obj (op X) := fun x =>
-  (ℱ'.cond _ (H.is_cover X)).amalgamate (pushforwardFamily α x) (pushforwardFamily_compatible H α x)
-#align category_theory.cover_dense.types.app_hom CategoryTheory.CoverDense.Types.appHom
+noncomputable def CategoryTheory.Functor.IsCoverDense.Types.appHom (X : D) :
+    ℱ.obj (op X) ⟶ ℱ'.val.obj (op X) := fun x =>
+  (ℱ'.cond _ (H.is_cover X)).amalgamate
+    (CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily α x)
+    (CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_compatible H α x)
+#align category_theory.cover_dense.types.app_hom CategoryTheory.Functor.IsCoverDense.Types.appHom
 -/
 
-#print CategoryTheory.CoverDense.Types.pushforwardFamily_apply /-
+#print CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_apply /-
 @[simp]
-theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y ⟶ X) :
-    pushforwardFamily α x f (Presieve.in_coverByImage G f) = α.app (op Y) (ℱ.map f.op x) :=
+theorem CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_apply {X} (x : ℱ.obj (op X))
+    {Y : C} (f : G.obj Y ⟶ X) :
+    CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily α x f
+        (Presieve.in_coverByImage G f) =
+      α.app (op Y) (ℱ.map f.op x) :=
   by
   unfold pushforward_family
   refine' congr_fun _ x
@@ -226,84 +238,92 @@ theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y 
   erw [← α.naturality (G.preimage _).op]
   simp only [← functor.map_comp, ← category.assoc, functor.comp_map, G.image_preimage, G.op_map,
     Quiver.Hom.unop_op, ← op_comp, presieve.cover_by_image_structure.fac]
-#align category_theory.cover_dense.types.pushforward_family_apply CategoryTheory.CoverDense.Types.pushforwardFamily_apply
+#align category_theory.cover_dense.types.pushforward_family_apply CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_apply
 -/
 
-#print CategoryTheory.CoverDense.Types.appHom_restrict /-
+#print CategoryTheory.Functor.IsCoverDense.Types.appHom_restrict /-
 @[simp]
-theorem appHom_restrict {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) (x) :
-    ℱ'.val.map f (appHom H α X x) = α.app (op Y) (ℱ.map f x) :=
+theorem CategoryTheory.Functor.IsCoverDense.Types.appHom_restrict {X : D} {Y : C}
+    (f : op X ⟶ op (G.obj Y)) (x) :
+    ℱ'.val.map f (CategoryTheory.Functor.IsCoverDense.Types.appHom H α X x) =
+      α.app (op Y) (ℱ.map f x) :=
   by
   refine'
     ((ℱ'.cond _ (H.is_cover X)).valid_glue (pushforward_family_compatible H α x) f.unop
           (presieve.in_cover_by_image G f.unop)).trans
       _
   apply pushforward_family_apply
-#align category_theory.cover_dense.types.app_hom_restrict CategoryTheory.CoverDense.Types.appHom_restrict
+#align category_theory.cover_dense.types.app_hom_restrict CategoryTheory.Functor.IsCoverDense.Types.appHom_restrict
 -/
 
-#print CategoryTheory.CoverDense.Types.appHom_valid_glue /-
+#print CategoryTheory.Functor.IsCoverDense.Types.appHom_valid_glue /-
 @[simp]
-theorem appHom_valid_glue {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) :
-    appHom H α X ≫ ℱ'.val.map f = ℱ.map f ≫ α.app (op Y) := by ext; apply app_hom_restrict
-#align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.CoverDense.Types.appHom_valid_glue
+theorem CategoryTheory.Functor.IsCoverDense.Types.appHom_valid_glue {X : D} {Y : C}
+    (f : op X ⟶ op (G.obj Y)) :
+    CategoryTheory.Functor.IsCoverDense.Types.appHom H α X ≫ ℱ'.val.map f =
+      ℱ.map f ≫ α.app (op Y) :=
+  by ext; apply app_hom_restrict
+#align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.Functor.IsCoverDense.Types.appHom_valid_glue
 -/
 
-#print CategoryTheory.CoverDense.Types.appIso /-
+#print CategoryTheory.Functor.IsCoverDense.Types.appIso /-
 /--
 (Implementation). The maps given in `app_iso` is inverse to each other and gives a `ℱ(X) ≅ ℱ'(X)`.
 -/
 @[simps]
-noncomputable def appIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) (X : D) :
-    ℱ.val.obj (op X) ≅ ℱ'.val.obj (op X)
+noncomputable def CategoryTheory.Functor.IsCoverDense.Types.appIso {ℱ ℱ' : SheafOfTypes.{v} K}
+    (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) (X : D) : ℱ.val.obj (op X) ≅ ℱ'.val.obj (op X)
     where
-  Hom := appHom H i.Hom X
-  inv := appHom H i.inv X
+  Hom := CategoryTheory.Functor.IsCoverDense.Types.appHom H i.Hom X
+  inv := CategoryTheory.Functor.IsCoverDense.Types.appHom H i.inv X
   hom_inv_id' := by ext x; apply H.ext; intro Y f; simp
   inv_hom_id' := by ext x; apply H.ext; intro Y f; simp
-#align category_theory.cover_dense.types.app_iso CategoryTheory.CoverDense.Types.appIso
+#align category_theory.cover_dense.types.app_iso CategoryTheory.Functor.IsCoverDense.Types.appIso
 -/
 
-#print CategoryTheory.CoverDense.Types.presheafHom /-
+#print CategoryTheory.Functor.IsCoverDense.Types.presheafHom /-
 /-- Given an natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of types, where `G` is full
 and cover-dense, and `ℱ'` is a sheaf, we may obtain a natural transformation between sheaves.
 -/
 @[simps]
-noncomputable def presheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ ⟶ ℱ'.val
+noncomputable def CategoryTheory.Functor.IsCoverDense.Types.presheafHom
+    (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ ⟶ ℱ'.val
     where
-  app X := appHom H α (unop X)
+  app X := CategoryTheory.Functor.IsCoverDense.Types.appHom H α (unop X)
   naturality' X Y f := by
     ext x
     apply H.ext ℱ' (unop Y)
     intro Y' f'
     simp only [app_hom_restrict, types_comp_apply, ← functor_to_types.map_comp_apply]
     rw [app_hom_restrict H α (f ≫ f'.op : op (unop X) ⟶ _)]
-#align category_theory.cover_dense.types.presheaf_hom CategoryTheory.CoverDense.Types.presheafHom
+#align category_theory.cover_dense.types.presheaf_hom CategoryTheory.Functor.IsCoverDense.Types.presheafHom
 -/
 
-#print CategoryTheory.CoverDense.Types.presheafIso /-
+#print CategoryTheory.Functor.IsCoverDense.Types.presheafIso /-
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of types, where `G` is full and
 cover-dense, and `ℱ, ℱ'` are sheaves, we may obtain a natural isomorphism between presheaves.
 -/
 @[simps]
-noncomputable def presheafIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) :
-    ℱ.val ≅ ℱ'.val :=
-  NatIso.ofComponents (fun X => appIso H i (unop X)) (presheafHom H i.Hom).naturality
-#align category_theory.cover_dense.types.presheaf_iso CategoryTheory.CoverDense.Types.presheafIso
+noncomputable def CategoryTheory.Functor.IsCoverDense.Types.presheafIso {ℱ ℱ' : SheafOfTypes.{v} K}
+    (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) : ℱ.val ≅ ℱ'.val :=
+  NatIso.ofComponents (fun X => CategoryTheory.Functor.IsCoverDense.Types.appIso H i (unop X))
+    (CategoryTheory.Functor.IsCoverDense.Types.presheafHom H i.Hom).naturality
+#align category_theory.cover_dense.types.presheaf_iso CategoryTheory.Functor.IsCoverDense.Types.presheafIso
 -/
 
-#print CategoryTheory.CoverDense.Types.sheafIso /-
+#print CategoryTheory.Functor.IsCoverDense.Types.sheafIso /-
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of types, where `G` is full and
 cover-dense, and `ℱ, ℱ'` are sheaves, we may obtain a natural isomorphism between sheaves.
 -/
 @[simps]
-noncomputable def sheafIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) : ℱ ≅ ℱ'
+noncomputable def CategoryTheory.Functor.IsCoverDense.Types.sheafIso {ℱ ℱ' : SheafOfTypes.{v} K}
+    (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) : ℱ ≅ ℱ'
     where
-  Hom := ⟨(presheafIso H i).Hom⟩
-  inv := ⟨(presheafIso H i).inv⟩
+  Hom := ⟨(CategoryTheory.Functor.IsCoverDense.Types.presheafIso H i).Hom⟩
+  inv := ⟨(CategoryTheory.Functor.IsCoverDense.Types.presheafIso H i).inv⟩
   hom_inv_id' := by ext1; apply (presheaf_iso H i).hom_inv_id
   inv_hom_id' := by ext1; apply (presheaf_iso H i).inv_hom_id
-#align category_theory.cover_dense.types.sheaf_iso CategoryTheory.CoverDense.Types.sheafIso
+#align category_theory.cover_dense.types.sheaf_iso CategoryTheory.Functor.IsCoverDense.Types.sheafIso
 -/
 
 end Types
@@ -312,14 +332,17 @@ open Types
 
 variable {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A}
 
-#print CategoryTheory.CoverDense.sheafCoyonedaHom /-
+#print CategoryTheory.Functor.IsCoverDense.sheafCoyonedaHom /-
 /-- (Implementation). The sheaf map given in `types.sheaf_hom` is natural in terms of `X`. -/
 @[simps]
-noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
+noncomputable def CategoryTheory.Functor.IsCoverDense.sheafCoyonedaHom
+    (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     coyoneda ⋙ (whiskeringLeft Dᵒᵖ A (Type _)).obj ℱ ⟶
       coyoneda ⋙ (whiskeringLeft Dᵒᵖ A (Type _)).obj ℱ'.val
     where
-  app X := presheafHom H (homOver α (unop X))
+  app X :=
+    CategoryTheory.Functor.IsCoverDense.Types.presheafHom H
+      (CategoryTheory.Functor.IsCoverDense.homOver α (unop X))
   naturality' X Y f := by
     ext U x
     change
@@ -338,14 +361,15 @@ noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     congr 1
     refine' (app_hom_restrict H (hom_over α (unop X)) hf'.some.map.op x).trans _
     simp
-#align category_theory.cover_dense.sheaf_coyoneda_hom CategoryTheory.CoverDense.sheafCoyonedaHom
+#align category_theory.cover_dense.sheaf_coyoneda_hom CategoryTheory.Functor.IsCoverDense.sheafCoyonedaHom
 -/
 
-#print CategoryTheory.CoverDense.sheafYonedaHom /-
+#print CategoryTheory.Functor.IsCoverDense.sheafYonedaHom /-
 /--
 (Implementation). `sheaf_coyoneda_hom` but the order of the arguments of the functor are swapped.
 -/
-noncomputable def sheafYonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ ⋙ yoneda ⟶ ℱ'.val ⋙ yoneda :=
+noncomputable def CategoryTheory.Functor.IsCoverDense.sheafYonedaHom
+    (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ ⋙ yoneda ⟶ ℱ'.val ⋙ yoneda :=
   by
   let α := sheaf_coyoneda_hom H α
   refine'
@@ -358,29 +382,30 @@ noncomputable def sheafYonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ
   · intro U V i
     ext X x
     exact congr_fun ((α.app X).naturality i) x
-#align category_theory.cover_dense.sheaf_yoneda_hom CategoryTheory.CoverDense.sheafYonedaHom
+#align category_theory.cover_dense.sheaf_yoneda_hom CategoryTheory.Functor.IsCoverDense.sheafYonedaHom
 -/
 
-#print CategoryTheory.CoverDense.sheafHom /-
+#print CategoryTheory.Functor.IsCoverDense.sheafHom /-
 /-- Given an natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ'` is a sheaf, we may obtain a natural transformation
 between presheaves.
 -/
-noncomputable def sheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ ⟶ ℱ'.val :=
-  let α' := sheafYonedaHom H α
+noncomputable def CategoryTheory.Functor.IsCoverDense.sheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
+    ℱ ⟶ ℱ'.val :=
+  let α' := CategoryTheory.Functor.IsCoverDense.sheafYonedaHom H α
   { app := fun X => yoneda.preimage (α'.app X)
     naturality' := fun X Y f => yoneda.map_injective (by simpa using α'.naturality f) }
-#align category_theory.cover_dense.sheaf_hom CategoryTheory.CoverDense.sheafHom
+#align category_theory.cover_dense.sheaf_hom CategoryTheory.Functor.IsCoverDense.sheafHom
 -/
 
-#print CategoryTheory.CoverDense.presheafIso /-
+#print CategoryTheory.Functor.IsCoverDense.presheafIso /-
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
 we may obtain a natural isomorphism between presheaves.
 -/
 @[simps]
-noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) :
-    ℱ.val ≅ ℱ'.val :=
+noncomputable def CategoryTheory.Functor.IsCoverDense.presheafIso {ℱ ℱ' : Sheaf K A}
+    (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) : ℱ.val ≅ ℱ'.val :=
   by
   have : ∀ X : Dᵒᵖ, is_iso ((sheaf_hom H i.hom).app X) :=
     by
@@ -394,28 +419,30 @@ noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G
     infer_instance
   haveI : is_iso (sheaf_hom H i.hom) := by apply nat_iso.is_iso_of_is_iso_app
   apply as_iso (sheaf_hom H i.hom)
-#align category_theory.cover_dense.presheaf_iso CategoryTheory.CoverDense.presheafIso
+#align category_theory.cover_dense.presheaf_iso CategoryTheory.Functor.IsCoverDense.presheafIso
 -/
 
-#print CategoryTheory.CoverDense.sheafIso /-
+#print CategoryTheory.Functor.IsCoverDense.sheafIso /-
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
 we may obtain a natural isomorphism between presheaves.
 -/
 @[simps]
-noncomputable def sheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) : ℱ ≅ ℱ'
+noncomputable def CategoryTheory.Functor.IsCoverDense.sheafIso {ℱ ℱ' : Sheaf K A}
+    (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) : ℱ ≅ ℱ'
     where
-  Hom := ⟨(presheafIso H i).Hom⟩
-  inv := ⟨(presheafIso H i).inv⟩
+  Hom := ⟨(CategoryTheory.Functor.IsCoverDense.presheafIso H i).Hom⟩
+  inv := ⟨(CategoryTheory.Functor.IsCoverDense.presheafIso H i).inv⟩
   hom_inv_id' := by ext1; apply (presheaf_iso H i).hom_inv_id
   inv_hom_id' := by ext1; apply (presheaf_iso H i).inv_hom_id
-#align category_theory.cover_dense.sheaf_iso CategoryTheory.CoverDense.sheafIso
+#align category_theory.cover_dense.sheaf_iso CategoryTheory.Functor.IsCoverDense.sheafIso
 -/
 
-#print CategoryTheory.CoverDense.sheafHom_restrict_eq /-
+#print CategoryTheory.Functor.IsCoverDense.sheafHom_restrict_eq /-
 /-- The constructed `sheaf_hom α` is equal to `α` when restricted onto `C`.
 -/
-theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : whiskerLeft G.op (sheafHom H α) = α :=
+theorem CategoryTheory.Functor.IsCoverDense.sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
+    whiskerLeft G.op (CategoryTheory.Functor.IsCoverDense.sheafHom H α) = α :=
   by
   ext X
   apply yoneda.map_injective
@@ -435,14 +462,15 @@ theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : whisker
   symm
   apply α.naturality (G.preimage hf.some.map).op
   infer_instance
-#align category_theory.cover_dense.sheaf_hom_restrict_eq CategoryTheory.CoverDense.sheafHom_restrict_eq
+#align category_theory.cover_dense.sheaf_hom_restrict_eq CategoryTheory.Functor.IsCoverDense.sheafHom_restrict_eq
 -/
 
-#print CategoryTheory.CoverDense.sheafHom_eq /-
+#print CategoryTheory.Functor.IsCoverDense.sheafHom_eq /-
 /-- If the pullback map is obtained via whiskering,
 then the result `sheaf_hom (whisker_left G.op α)` is equal to `α`.
 -/
-theorem sheafHom_eq (α : ℱ ⟶ ℱ'.val) : sheafHom H (whiskerLeft G.op α) = α :=
+theorem CategoryTheory.Functor.IsCoverDense.sheafHom_eq (α : ℱ ⟶ ℱ'.val) :
+    CategoryTheory.Functor.IsCoverDense.sheafHom H (whiskerLeft G.op α) = α :=
   by
   ext X
   apply yoneda.map_injective
@@ -456,39 +484,40 @@ theorem sheafHom_eq (α : ℱ ⟶ ℱ'.val) : sheafHom H (whiskerLeft G.op α) =
   conv_lhs => rw [← hf.some.fac]
   dsimp
   simp
-#align category_theory.cover_dense.sheaf_hom_eq CategoryTheory.CoverDense.sheafHom_eq
+#align category_theory.cover_dense.sheaf_hom_eq CategoryTheory.Functor.IsCoverDense.sheafHom_eq
 -/
 
-#print CategoryTheory.CoverDense.restrictHomEquivHom /-
+#print CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom /-
 /-- A full and cover-dense functor `G` induces an equivalence between morphisms into a sheaf and
 morphisms over the restrictions via `G`.
 -/
-noncomputable def restrictHomEquivHom : (G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) ≃ (ℱ ⟶ ℱ'.val)
+noncomputable def CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom :
+    (G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) ≃ (ℱ ⟶ ℱ'.val)
     where
-  toFun := sheafHom H
+  toFun := CategoryTheory.Functor.IsCoverDense.sheafHom H
   invFun := whiskerLeft G.op
-  left_inv := sheafHom_restrict_eq H
-  right_inv := sheafHom_eq H
-#align category_theory.cover_dense.restrict_hom_equiv_hom CategoryTheory.CoverDense.restrictHomEquivHom
+  left_inv := CategoryTheory.Functor.IsCoverDense.sheafHom_restrict_eq H
+  right_inv := CategoryTheory.Functor.IsCoverDense.sheafHom_eq H
+#align category_theory.cover_dense.restrict_hom_equiv_hom CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom
 -/
 
-#print CategoryTheory.CoverDense.iso_of_restrict_iso /-
+#print CategoryTheory.Functor.IsCoverDense.iso_of_restrict_iso /-
 /-- Given a full and cover-dense functor `G` and a natural transformation of sheaves `α : ℱ ⟶ ℱ'`,
 if the pullback of `α` along `G` is iso, then `α` is also iso.
 -/
-theorem iso_of_restrict_iso {ℱ ℱ' : Sheaf K A} (α : ℱ ⟶ ℱ') (i : IsIso (whiskerLeft G.op α.val)) :
-    IsIso α :=
+theorem CategoryTheory.Functor.IsCoverDense.iso_of_restrict_iso {ℱ ℱ' : Sheaf K A} (α : ℱ ⟶ ℱ')
+    (i : IsIso (whiskerLeft G.op α.val)) : IsIso α :=
   by
   convert is_iso.of_iso (sheaf_iso H (as_iso (whisker_left G.op α.val))) using 1
   ext1
   apply (sheaf_hom_eq _ _).symm
-#align category_theory.cover_dense.iso_of_restrict_iso CategoryTheory.CoverDense.iso_of_restrict_iso
+#align category_theory.cover_dense.iso_of_restrict_iso CategoryTheory.Functor.IsCoverDense.iso_of_restrict_iso
 -/
 
-#print CategoryTheory.CoverDense.compatiblePreserving /-
+#print CategoryTheory.Functor.IsCoverDense.compatiblePreserving /-
 /-- A fully faithful cover-dense functor preserves compatible families. -/
-theorem compatiblePreserving [Faithful G] : CompatiblePreserving K G :=
-  by
+theorem CategoryTheory.Functor.IsCoverDense.compatiblePreserving [Faithful G] :
+    CompatiblePreserving K G := by
   constructor
   intro ℱ Z T x hx Y₁ Y₂ X f₁ f₂ g₁ g₂ hg₁ hg₂ eq
   apply H.ext
@@ -499,35 +528,37 @@ theorem compatiblePreserving [Faithful G] : CompatiblePreserving K G :=
   apply hx
   apply G.map_injective
   simp [Eq]
-#align category_theory.cover_dense.compatible_preserving CategoryTheory.CoverDense.compatiblePreserving
+#align category_theory.cover_dense.compatible_preserving CategoryTheory.Functor.IsCoverDense.compatiblePreserving
 -/
 
-#print CategoryTheory.CoverDense.Sites.Pullback.full /-
-noncomputable instance Sites.Pullback.full [Faithful G] (Hp : CoverPreserving J K G) :
-    Full (Sites.pullback A H.CompatiblePreserving Hp)
+#print CategoryTheory.Functor.IsCoverDense.fullSheafPushforwardContinuous /-
+noncomputable instance CategoryTheory.Functor.IsCoverDense.fullSheafPushforwardContinuous
+    [Faithful G] (Hp : CoverPreserving J K G) :
+    Full (Functor.sheafPushforwardContinuous A H.CompatiblePreserving Hp)
     where
   preimage ℱ ℱ' α := ⟨H.sheafHom α.val⟩
   witness' ℱ ℱ' α := Sheaf.Hom.ext _ _ <| H.sheafHom_restrict_eq α.val
-#align category_theory.cover_dense.sites.pullback.full CategoryTheory.CoverDense.Sites.Pullback.full
+#align category_theory.cover_dense.sites.pullback.full CategoryTheory.Functor.IsCoverDense.fullSheafPushforwardContinuous
 -/
 
-#print CategoryTheory.CoverDense.Sites.Pullback.faithful /-
-instance Sites.Pullback.faithful [Faithful G] (Hp : CoverPreserving J K G) :
-    Faithful (Sites.pullback A H.CompatiblePreserving Hp)
+#print CategoryTheory.Functor.IsCoverDense.faithful_sheafPushforwardContinuous /-
+instance CategoryTheory.Functor.IsCoverDense.faithful_sheafPushforwardContinuous [Faithful G]
+    (Hp : CoverPreserving J K G) :
+    Faithful (Functor.sheafPushforwardContinuous A H.CompatiblePreserving Hp)
     where map_injective' := by
     intro ℱ ℱ' α β e
     ext1
     apply_fun fun e => e.val at e 
     dsimp at e 
     rw [← H.sheaf_hom_eq α.val, ← H.sheaf_hom_eq β.val, e]
-#align category_theory.cover_dense.sites.pullback.faithful CategoryTheory.CoverDense.Sites.Pullback.faithful
+#align category_theory.cover_dense.sites.pullback.faithful CategoryTheory.Functor.IsCoverDense.faithful_sheafPushforwardContinuous
 -/
 
 end CoverDense
 
 end CategoryTheory
 
-namespace CategoryTheory.CoverDense
+namespace CategoryTheory.Functor.IsCoverDense
 
 open CategoryTheory
 
@@ -539,17 +570,18 @@ variable {J : GrothendieckTopology C} {K : GrothendieckTopology D}
 
 variable {A : Type w} [Category.{max u v} A] [Limits.HasLimits A]
 
-variable (Hd : CoverDense K G) (Hp : CoverPreserving J K G) (Hl : CoverLifting J K G)
+variable (Hd : CategoryTheory.Functor.IsCoverDense K G) (Hp : CoverPreserving J K G)
+  (Hl : CategoryTheory.Functor.IsCocontinuous J K G)
 
-#print CategoryTheory.CoverDense.sheafEquivOfCoverPreservingCoverLifting /-
+#print CategoryTheory.Functor.IsCoverDense.sheafEquivOfCoverPreservingCoverLifting /-
 /-- Given a functor between small sites that is cover-dense, cover-preserving, and cover-lifting,
 it induces an equivalence of category of sheaves valued in a complete category.
 -/
 @[simps Functor inverse]
-noncomputable def sheafEquivOfCoverPreservingCoverLifting : Sheaf J A ≌ Sheaf K A :=
-  by
+noncomputable def CategoryTheory.Functor.IsCoverDense.sheafEquivOfCoverPreservingCoverLifting :
+    Sheaf J A ≌ Sheaf K A := by
   symm
-  let α := Sites.pullbackCopullbackAdjunction.{w, v, u} A Hp Hl Hd.compatible_preserving
+  let α := Functor.sheafAdjunctionCocontinuous.{w, v, u} A Hp Hl Hd.compatible_preserving
   have : ∀ X : Sheaf J A, is_iso (α.counit.app X) :=
     by
     intro ℱ
@@ -562,8 +594,8 @@ noncomputable def sheafEquivOfCoverPreservingCoverLifting : Sheaf J A ≌ Sheaf
       unitIso := as_iso α.unit
       counitIso := as_iso α.counit
       functor_unitIso_comp' := fun ℱ => by convert α.left_triangle_components }
-#align category_theory.cover_dense.Sheaf_equiv_of_cover_preserving_cover_lifting CategoryTheory.CoverDense.sheafEquivOfCoverPreservingCoverLifting
+#align category_theory.cover_dense.Sheaf_equiv_of_cover_preserving_cover_lifting CategoryTheory.Functor.IsCoverDense.sheafEquivOfCoverPreservingCoverLifting
 -/
 
-end CategoryTheory.CoverDense
+end CategoryTheory.Functor.IsCoverDense
 

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.Sites.Sheaf
-import Mathbin.CategoryTheory.Sites.CoverLifting
-import Mathbin.CategoryTheory.Adjunction.FullyFaithful
+import CategoryTheory.Sites.Sheaf
+import CategoryTheory.Sites.CoverLifting
+import CategoryTheory.Adjunction.FullyFaithful
 
 #align_import category_theory.sites.dense_subsite from "leanprover-community/mathlib"@"2ed2c6310e6f1c5562bdf6bfbda55ebbf6891abe"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -68,8 +68,6 @@ structure Presieve.CoverByImageStructure (G : C ⥤ D) {V U : D} (f : V ⟶ U) w
 #align category_theory.presieve.cover_by_image_structure CategoryTheory.Presieve.CoverByImageStructure
 -/
 
-restate_axiom presieve.cover_by_image_structure.fac'
-
 attribute [simp, reassoc] presieve.cover_by_image_structure.fac
 
 #print CategoryTheory.Presieve.coverByImage /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

@@ -388,7 +388,7 @@ noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G
     by
     intro X
     apply is_iso_of_reflects_iso _ yoneda
-    use (sheaf_yoneda_hom H i.inv).app X
+    use(sheaf_yoneda_hom H i.inv).app X
     constructor <;> ext x : 2 <;>
       simp only [sheaf_hom, nat_trans.comp_app, nat_trans.id_app, functor.image_preimage]
     exact ((presheaf_iso H (iso_over i (unop x))).app X).hom_inv_id

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.dense_subsite
-! leanprover-community/mathlib commit 2ed2c6310e6f1c5562bdf6bfbda55ebbf6891abe
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.CategoryTheory.Sites.Sheaf
 import Mathbin.CategoryTheory.Sites.CoverLifting
 import Mathbin.CategoryTheory.Adjunction.FullyFaithful
 
+#align_import category_theory.sites.dense_subsite from "leanprover-community/mathlib"@"2ed2c6310e6f1c5562bdf6bfbda55ebbf6891abe"
+
 /-!
 # Dense subsites
 

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

@@ -326,7 +326,7 @@ noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     where
   app X := presheafHom H (homOver α (unop X))
   naturality' X Y f := by
-    ext (U x)
+    ext U x
     change
       app_hom H (hom_over α (unop Y)) (unop U) (f.unop ≫ x) =
         f.unop ≫ app_hom H (hom_over α (unop X)) (unop U) x
@@ -361,7 +361,7 @@ noncomputable def sheafYonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ
       { app := fun X => (α.app X).app U
         naturality' := fun X Y f => by simpa using congr_app (α.naturality f) U }
   · intro U V i
-    ext (X x)
+    ext X x
     exact congr_fun ((α.app X).naturality i) x
 #align category_theory.cover_dense.sheaf_yoneda_hom CategoryTheory.CoverDense.sheafYonedaHom
 -/

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

@@ -94,10 +94,12 @@ def Sieve.coverByImage (G : C ⥤ D) (U : D) : Sieve U :=
 #align category_theory.sieve.cover_by_image CategoryTheory.Sieve.coverByImage
 -/
 
+#print CategoryTheory.Presieve.in_coverByImage /-
 theorem Presieve.in_coverByImage (G : C ⥤ D) {X : D} {Y : C} (f : G.obj Y ⟶ X) :
     Presieve.coverByImage G X f :=
   ⟨⟨Y, 𝟙 _, f, by simp⟩⟩
 #align category_theory.presieve.in_cover_by_image CategoryTheory.Presieve.in_coverByImage
+-/
 
 #print CategoryTheory.CoverDense /-
 /-- A functor `G : (C, J) ⥤ (D, K)` is called `cover_dense` if for each object in `D`,
@@ -118,6 +120,7 @@ variable {K}
 
 variable {A : Type _} [Category A] {G : C ⥤ D} (H : CoverDense K G)
 
+#print CategoryTheory.CoverDense.ext /-
 -- this is not marked with `@[ext]` because `H` can not be inferred from the type
 theorem ext (H : CoverDense K G) (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.obj (op X)}
     (h : ∀ ⦃Y : C⦄ (f : G.obj Y ⟶ X), ℱ.val.map f.op s = ℱ.val.map f.op t) : s = t :=
@@ -126,7 +129,9 @@ theorem ext (H : CoverDense K G) (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.o
   rintro Y _ ⟨Z, f₁, f₂, ⟨rfl⟩⟩
   simp [h f₂]
 #align category_theory.cover_dense.ext CategoryTheory.CoverDense.ext
+-/
 
+#print CategoryTheory.CoverDense.functorPullback_pushforward_covering /-
 theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X : C}
     (T : K (G.obj X)) : (T.val.functorPullback G).functorPushforward G ∈ K (G.obj X) :=
   by
@@ -137,7 +142,9 @@ theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X :
   · simpa using T.val.downward_closed hf f'
   · simp
 #align category_theory.cover_dense.functor_pullback_pushforward_covering CategoryTheory.CoverDense.functorPullback_pushforward_covering
+-/
 
+#print CategoryTheory.CoverDense.homOver /-
 /-- (Implementation). Given an hom between the pullbacks of two sheaves, we can whisker it with
 `coyoneda` to obtain an hom between the pullbacks of the sheaves of maps from `X`.
 -/
@@ -146,6 +153,7 @@ def homOver {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A} (α : G.op ⋙ ℱ ⟶ G.op
     G.op ⋙ ℱ ⋙ coyoneda.obj (op X) ⟶ G.op ⋙ (sheafOver ℱ' X).val :=
   whiskerRight α (coyoneda.obj (op X))
 #align category_theory.cover_dense.hom_over CategoryTheory.CoverDense.homOver
+-/
 
 #print CategoryTheory.CoverDense.isoOver /-
 /-- (Implementation). Given an iso between the pullbacks of two sheaves, we can whisker it with
@@ -158,11 +166,13 @@ def isoOver {ℱ ℱ' : Sheaf K A} (α : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val)
 #align category_theory.cover_dense.iso_over CategoryTheory.CoverDense.isoOver
 -/
 
+#print CategoryTheory.CoverDense.sheaf_eq_amalgamation /-
 theorem sheaf_eq_amalgamation (ℱ : Sheaf K A) {X : A} {U : D} {T : Sieve U} (hT)
     (x : FamilyOfElements _ T) (hx) (t) (h : x.IsAmalgamation t) :
     t = (ℱ.cond X T hT).amalgamate x hx :=
   (ℱ.cond X T hT).IsSeparatedFor x t _ h ((ℱ.cond X T hT).IsAmalgamation hx)
 #align category_theory.cover_dense.sheaf_eq_amalgamation CategoryTheory.CoverDense.sheaf_eq_amalgamation
+-/
 
 variable [Full G]
 
@@ -170,6 +180,7 @@ namespace Types
 
 variable {ℱ : Dᵒᵖ ⥤ Type v} {ℱ' : SheafOfTypes.{v} K} (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val)
 
+#print CategoryTheory.CoverDense.Types.pushforwardFamily /-
 /--
 (Implementation). Given a section of `ℱ` on `X`, we can obtain a family of elements valued in `ℱ'`
 that is defined on a cover generated by the images of `G`. -/
@@ -178,9 +189,9 @@ noncomputable def pushforwardFamily {X} (x : ℱ.obj (op X)) :
     FamilyOfElements ℱ'.val (coverByImage G X) := fun Y f hf =>
   ℱ'.val.map hf.some.lift.op <| α.app (op _) (ℱ.map hf.some.map.op x : _)
 #align category_theory.cover_dense.types.pushforward_family CategoryTheory.CoverDense.Types.pushforwardFamily
+-/
 
-include H
-
+#print CategoryTheory.CoverDense.Types.pushforwardFamily_compatible /-
 /-- (Implementation). The `pushforward_family` defined is compatible. -/
 theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) : (pushforwardFamily α x).Compatible :=
   by
@@ -199,14 +210,16 @@ theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) : (pushforwardFami
   congr 3
   simp [e]
 #align category_theory.cover_dense.types.pushforward_family_compatible CategoryTheory.CoverDense.Types.pushforwardFamily_compatible
+-/
 
-omit H
-
+#print CategoryTheory.CoverDense.Types.appHom /-
 /-- (Implementation). The morphism `ℱ(X) ⟶ ℱ'(X)` given by gluing the `pushforward_family`. -/
 noncomputable def appHom (X : D) : ℱ.obj (op X) ⟶ ℱ'.val.obj (op X) := fun x =>
   (ℱ'.cond _ (H.is_cover X)).amalgamate (pushforwardFamily α x) (pushforwardFamily_compatible H α x)
 #align category_theory.cover_dense.types.app_hom CategoryTheory.CoverDense.Types.appHom
+-/
 
+#print CategoryTheory.CoverDense.Types.pushforwardFamily_apply /-
 @[simp]
 theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y ⟶ X) :
     pushforwardFamily α x f (Presieve.in_coverByImage G f) = α.app (op Y) (ℱ.map f.op x) :=
@@ -219,7 +232,9 @@ theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y 
   simp only [← functor.map_comp, ← category.assoc, functor.comp_map, G.image_preimage, G.op_map,
     Quiver.Hom.unop_op, ← op_comp, presieve.cover_by_image_structure.fac]
 #align category_theory.cover_dense.types.pushforward_family_apply CategoryTheory.CoverDense.Types.pushforwardFamily_apply
+-/
 
+#print CategoryTheory.CoverDense.Types.appHom_restrict /-
 @[simp]
 theorem appHom_restrict {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) (x) :
     ℱ'.val.map f (appHom H α X x) = α.app (op Y) (ℱ.map f x) :=
@@ -230,12 +245,16 @@ theorem appHom_restrict {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) (x) :
       _
   apply pushforward_family_apply
 #align category_theory.cover_dense.types.app_hom_restrict CategoryTheory.CoverDense.Types.appHom_restrict
+-/
 
+#print CategoryTheory.CoverDense.Types.appHom_valid_glue /-
 @[simp]
 theorem appHom_valid_glue {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) :
     appHom H α X ≫ ℱ'.val.map f = ℱ.map f ≫ α.app (op Y) := by ext; apply app_hom_restrict
 #align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.CoverDense.Types.appHom_valid_glue
+-/
 
+#print CategoryTheory.CoverDense.Types.appIso /-
 /--
 (Implementation). The maps given in `app_iso` is inverse to each other and gives a `ℱ(X) ≅ ℱ'(X)`.
 -/
@@ -248,6 +267,7 @@ noncomputable def appIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val 
   hom_inv_id' := by ext x; apply H.ext; intro Y f; simp
   inv_hom_id' := by ext x; apply H.ext; intro Y f; simp
 #align category_theory.cover_dense.types.app_iso CategoryTheory.CoverDense.Types.appIso
+-/
 
 #print CategoryTheory.CoverDense.Types.presheafHom /-
 /-- Given an natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of types, where `G` is full
@@ -297,6 +317,7 @@ open Types
 
 variable {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A}
 
+#print CategoryTheory.CoverDense.sheafCoyonedaHom /-
 /-- (Implementation). The sheaf map given in `types.sheaf_hom` is natural in terms of `X`. -/
 @[simps]
 noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
@@ -323,8 +344,7 @@ noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     refine' (app_hom_restrict H (hom_over α (unop X)) hf'.some.map.op x).trans _
     simp
 #align category_theory.cover_dense.sheaf_coyoneda_hom CategoryTheory.CoverDense.sheafCoyonedaHom
-
-include H
+-/
 
 #print CategoryTheory.CoverDense.sheafYonedaHom /-
 /--
@@ -346,8 +366,6 @@ noncomputable def sheafYonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ
 #align category_theory.cover_dense.sheaf_yoneda_hom CategoryTheory.CoverDense.sheafYonedaHom
 -/
 
-omit H
-
 #print CategoryTheory.CoverDense.sheafHom /-
 /-- Given an natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ'` is a sheaf, we may obtain a natural transformation
@@ -360,8 +378,6 @@ noncomputable def sheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ ⟶ 
 #align category_theory.cover_dense.sheaf_hom CategoryTheory.CoverDense.sheafHom
 -/
 
-include H
-
 #print CategoryTheory.CoverDense.presheafIso /-
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
@@ -386,8 +402,6 @@ noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G
 #align category_theory.cover_dense.presheaf_iso CategoryTheory.CoverDense.presheafIso
 -/
 
-omit H
-
 #print CategoryTheory.CoverDense.sheafIso /-
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
@@ -403,6 +417,7 @@ noncomputable def sheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G.op
 #align category_theory.cover_dense.sheaf_iso CategoryTheory.CoverDense.sheafIso
 -/
 
+#print CategoryTheory.CoverDense.sheafHom_restrict_eq /-
 /-- The constructed `sheaf_hom α` is equal to `α` when restricted onto `C`.
 -/
 theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : whiskerLeft G.op (sheafHom H α) = α :=
@@ -426,7 +441,9 @@ theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : whisker
   apply α.naturality (G.preimage hf.some.map).op
   infer_instance
 #align category_theory.cover_dense.sheaf_hom_restrict_eq CategoryTheory.CoverDense.sheafHom_restrict_eq
+-/
 
+#print CategoryTheory.CoverDense.sheafHom_eq /-
 /-- If the pullback map is obtained via whiskering,
 then the result `sheaf_hom (whisker_left G.op α)` is equal to `α`.
 -/
@@ -445,6 +462,7 @@ theorem sheafHom_eq (α : ℱ ⟶ ℱ'.val) : sheafHom H (whiskerLeft G.op α) =
   dsimp
   simp
 #align category_theory.cover_dense.sheaf_hom_eq CategoryTheory.CoverDense.sheafHom_eq
+-/
 
 #print CategoryTheory.CoverDense.restrictHomEquivHom /-
 /-- A full and cover-dense functor `G` induces an equivalence between morphisms into a sheaf and
@@ -459,8 +477,7 @@ noncomputable def restrictHomEquivHom : (G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) ≃
 #align category_theory.cover_dense.restrict_hom_equiv_hom CategoryTheory.CoverDense.restrictHomEquivHom
 -/
 
-include H
-
+#print CategoryTheory.CoverDense.iso_of_restrict_iso /-
 /-- Given a full and cover-dense functor `G` and a natural transformation of sheaves `α : ℱ ⟶ ℱ'`,
 if the pullback of `α` along `G` is iso, then `α` is also iso.
 -/
@@ -471,7 +488,9 @@ theorem iso_of_restrict_iso {ℱ ℱ' : Sheaf K A} (α : ℱ ⟶ ℱ') (i : IsIs
   ext1
   apply (sheaf_hom_eq _ _).symm
 #align category_theory.cover_dense.iso_of_restrict_iso CategoryTheory.CoverDense.iso_of_restrict_iso
+-/
 
+#print CategoryTheory.CoverDense.compatiblePreserving /-
 /-- A fully faithful cover-dense functor preserves compatible families. -/
 theorem compatiblePreserving [Faithful G] : CompatiblePreserving K G :=
   by
@@ -486,8 +505,7 @@ theorem compatiblePreserving [Faithful G] : CompatiblePreserving K G :=
   apply G.map_injective
   simp [Eq]
 #align category_theory.cover_dense.compatible_preserving CategoryTheory.CoverDense.compatiblePreserving
-
-omit H
+-/
 
 #print CategoryTheory.CoverDense.Sites.Pullback.full /-
 noncomputable instance Sites.Pullback.full [Faithful G] (Hp : CoverPreserving J K G) :
@@ -528,8 +546,7 @@ variable {A : Type w} [Category.{max u v} A] [Limits.HasLimits A]
 
 variable (Hd : CoverDense K G) (Hp : CoverPreserving J K G) (Hl : CoverLifting J K G)
 
-include Hd Hp Hl
-
+#print CategoryTheory.CoverDense.sheafEquivOfCoverPreservingCoverLifting /-
 /-- Given a functor between small sites that is cover-dense, cover-preserving, and cover-lifting,
 it induces an equivalence of category of sheaves valued in a complete category.
 -/
@@ -551,6 +568,7 @@ noncomputable def sheafEquivOfCoverPreservingCoverLifting : Sheaf J A ≌ Sheaf
       counitIso := as_iso α.counit
       functor_unitIso_comp' := fun ℱ => by convert α.left_triangle_components }
 #align category_theory.cover_dense.Sheaf_equiv_of_cover_preserving_cover_lifting CategoryTheory.CoverDense.sheafEquivOfCoverPreservingCoverLifting
+-/
 
 end CategoryTheory.CoverDense
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -504,7 +504,7 @@ instance Sites.Pullback.faithful [Faithful G] (Hp : CoverPreserving J K G) :
     where map_injective' := by
     intro ℱ ℱ' α β e
     ext1
-    apply_fun fun e => e.val  at e 
+    apply_fun fun e => e.val at e 
     dsimp at e 
     rw [← H.sheaf_hom_eq α.val, ← H.sheaf_hom_eq β.val, e]
 #align category_theory.cover_dense.sites.pullback.faithful CategoryTheory.CoverDense.Sites.Pullback.faithful

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

@@ -313,7 +313,7 @@ noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     apply sheaf_eq_amalgamation
     apply H.is_cover
     intro Y' f' hf'
-    change unop X ⟶ ℱ.obj (op (unop _)) at x
+    change unop X ⟶ ℱ.obj (op (unop _)) at x 
     dsimp
     simp only [pushforward_family, functor.comp_map, coyoneda_obj_map, hom_over_app, category.assoc]
     congr 1
@@ -504,8 +504,8 @@ instance Sites.Pullback.faithful [Faithful G] (Hp : CoverPreserving J K G) :
     where map_injective' := by
     intro ℱ ℱ' α β e
     ext1
-    apply_fun fun e => e.val  at e
-    dsimp at e
+    apply_fun fun e => e.val  at e 
+    dsimp at e 
     rw [← H.sheaf_hom_eq α.val, ← H.sheaf_hom_eq β.val, e]
 #align category_theory.cover_dense.sites.pullback.faithful CategoryTheory.CoverDense.Sites.Pullback.faithful
 -/

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

@@ -94,12 +94,6 @@ def Sieve.coverByImage (G : C ⥤ D) (U : D) : Sieve U :=
 #align category_theory.sieve.cover_by_image CategoryTheory.Sieve.coverByImage
 -/
 
-/- warning: category_theory.presieve.in_cover_by_image -> CategoryTheory.Presieve.in_coverByImage is a dubious translation:
-lean 3 declaration is
-  forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {D : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u3} D] (G : CategoryTheory.Functor.{u2, u4, u1, u3} C _inst_1 D _inst_2) {X : D} {Y : C} (f : Quiver.Hom.{succ u4, u3} D (CategoryTheory.CategoryStruct.toQuiver.{u4, u3} D (CategoryTheory.Category.toCategoryStruct.{u4, u3} D _inst_2)) (CategoryTheory.Functor.obj.{u2, u4, u1, u3} C _inst_1 D _inst_2 G Y) X), CategoryTheory.Presieve.coverByImage.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X (CategoryTheory.Functor.obj.{u2, u4, u1, u3} C _inst_1 D _inst_2 G Y) f
-but is expected to have type
-  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u4, u2} C] {D : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u3, u1} D] (G : CategoryTheory.Functor.{u4, u3, u2, u1} C _inst_1 D _inst_2) {X : D} {Y : C} (f : Quiver.Hom.{succ u3, u1} D (CategoryTheory.CategoryStruct.toQuiver.{u3, u1} D (CategoryTheory.Category.toCategoryStruct.{u3, u1} D _inst_2)) (Prefunctor.obj.{succ u4, succ u3, u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} C (CategoryTheory.Category.toCategoryStruct.{u4, u2} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u1} D (CategoryTheory.Category.toCategoryStruct.{u3, u1} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u4, u3, u2, u1} C _inst_1 D _inst_2 G) Y) X), CategoryTheory.Presieve.coverByImage.{u2, u4, u1, u3} C _inst_1 D _inst_2 G X (Prefunctor.obj.{succ u4, succ u3, u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u4, u2} C (CategoryTheory.Category.toCategoryStruct.{u4, u2} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u3, u1} D (CategoryTheory.Category.toCategoryStruct.{u3, u1} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u4, u3, u2, u1} C _inst_1 D _inst_2 G) Y) f
-Case conversion may be inaccurate. Consider using '#align category_theory.presieve.in_cover_by_image CategoryTheory.Presieve.in_coverByImageₓ'. -/
 theorem Presieve.in_coverByImage (G : C ⥤ D) {X : D} {Y : C} (f : G.obj Y ⟶ X) :
     Presieve.coverByImage G X f :=
   ⟨⟨Y, 𝟙 _, f, by simp⟩⟩
@@ -124,9 +118,6 @@ variable {K}
 
 variable {A : Type _} [Category A] {G : C ⥤ D} (H : CoverDense K G)
 
-/- warning: category_theory.cover_dense.ext -> CategoryTheory.CoverDense.ext is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.ext CategoryTheory.CoverDense.extₓ'. -/
 -- this is not marked with `@[ext]` because `H` can not be inferred from the type
 theorem ext (H : CoverDense K G) (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.obj (op X)}
     (h : ∀ ⦃Y : C⦄ (f : G.obj Y ⟶ X), ℱ.val.map f.op s = ℱ.val.map f.op t) : s = t :=
@@ -136,9 +127,6 @@ theorem ext (H : CoverDense K G) (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.o
   simp [h f₂]
 #align category_theory.cover_dense.ext CategoryTheory.CoverDense.ext
 
-/- warning: category_theory.cover_dense.functor_pullback_pushforward_covering -> CategoryTheory.CoverDense.functorPullback_pushforward_covering is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.functor_pullback_pushforward_covering CategoryTheory.CoverDense.functorPullback_pushforward_coveringₓ'. -/
 theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X : C}
     (T : K (G.obj X)) : (T.val.functorPullback G).functorPushforward G ∈ K (G.obj X) :=
   by
@@ -150,12 +138,6 @@ theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X :
   · simp
 #align category_theory.cover_dense.functor_pullback_pushforward_covering CategoryTheory.CoverDense.functorPullback_pushforward_covering
 
-/- warning: category_theory.cover_dense.hom_over -> CategoryTheory.CoverDense.homOver is a dubious translation:
-lean 3 declaration is
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 /-- (Implementation). Given an hom between the pullbacks of two sheaves, we can whisker it with
 `coyoneda` to obtain an hom between the pullbacks of the sheaves of maps from `X`.
 -/
@@ -176,9 +158,6 @@ def isoOver {ℱ ℱ' : Sheaf K A} (α : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val)
 #align category_theory.cover_dense.iso_over CategoryTheory.CoverDense.isoOver
 -/
 
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 theorem sheaf_eq_amalgamation (ℱ : Sheaf K A) {X : A} {U : D} {T : Sieve U} (hT)
     (x : FamilyOfElements _ T) (hx) (t) (h : x.IsAmalgamation t) :
     t = (ℱ.cond X T hT).amalgamate x hx :=
@@ -191,12 +170,6 @@ namespace Types
 
 variable {ℱ : Dᵒᵖ ⥤ Type v} {ℱ' : SheafOfTypes.{v} K} (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val)
 
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 /--
 (Implementation). Given a section of `ℱ` on `X`, we can obtain a family of elements valued in `ℱ'`
 that is defined on a cover generated by the images of `G`. -/
@@ -208,12 +181,6 @@ noncomputable def pushforwardFamily {X} (x : ℱ.obj (op X)) :
 
 include H
 
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 /-- (Implementation). The `pushforward_family` defined is compatible. -/
 theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) : (pushforwardFamily α x).Compatible :=
   by
@@ -235,20 +202,11 @@ theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) : (pushforwardFami
 
 omit H
 
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 /-- (Implementation). The morphism `ℱ(X) ⟶ ℱ'(X)` given by gluing the `pushforward_family`. -/
 noncomputable def appHom (X : D) : ℱ.obj (op X) ⟶ ℱ'.val.obj (op X) := fun x =>
   (ℱ'.cond _ (H.is_cover X)).amalgamate (pushforwardFamily α x) (pushforwardFamily_compatible H α x)
 #align category_theory.cover_dense.types.app_hom CategoryTheory.CoverDense.Types.appHom
 
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 @[simp]
 theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y ⟶ X) :
     pushforwardFamily α x f (Presieve.in_coverByImage G f) = α.app (op Y) (ℱ.map f.op x) :=
@@ -262,9 +220,6 @@ theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y 
     Quiver.Hom.unop_op, ← op_comp, presieve.cover_by_image_structure.fac]
 #align category_theory.cover_dense.types.pushforward_family_apply CategoryTheory.CoverDense.Types.pushforwardFamily_apply
 
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 @[simp]
 theorem appHom_restrict {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) (x) :
     ℱ'.val.map f (appHom H α X x) = α.app (op Y) (ℱ.map f x) :=
@@ -276,20 +231,11 @@ theorem appHom_restrict {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) (x) :
   apply pushforward_family_apply
 #align category_theory.cover_dense.types.app_hom_restrict CategoryTheory.CoverDense.Types.appHom_restrict
 
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 @[simp]
 theorem appHom_valid_glue {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) :
     appHom H α X ≫ ℱ'.val.map f = ℱ.map f ≫ α.app (op Y) := by ext; apply app_hom_restrict
 #align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.CoverDense.Types.appHom_valid_glue
 
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 /--
 (Implementation). The maps given in `app_iso` is inverse to each other and gives a `ℱ(X) ≅ ℱ'(X)`.
 -/
@@ -351,9 +297,6 @@ open Types
 
 variable {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A}
 
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 /-- (Implementation). The sheaf map given in `types.sheaf_hom` is natural in terms of `X`. -/
 @[simps]
 noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
@@ -460,12 +403,6 @@ noncomputable def sheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G.op
 #align category_theory.cover_dense.sheaf_iso CategoryTheory.CoverDense.sheafIso
 -/
 
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 /-- The constructed `sheaf_hom α` is equal to `α` when restricted onto `C`.
 -/
 theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : whiskerLeft G.op (sheafHom H α) = α :=
@@ -490,12 +427,6 @@ theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : whisker
   infer_instance
 #align category_theory.cover_dense.sheaf_hom_restrict_eq CategoryTheory.CoverDense.sheafHom_restrict_eq
 
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 /-- If the pullback map is obtained via whiskering,
 then the result `sheaf_hom (whisker_left G.op α)` is equal to `α`.
 -/
@@ -530,12 +461,6 @@ noncomputable def restrictHomEquivHom : (G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) ≃
 
 include H
 
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 /-- Given a full and cover-dense functor `G` and a natural transformation of sheaves `α : ℱ ⟶ ℱ'`,
 if the pullback of `α` along `G` is iso, then `α` is also iso.
 -/
@@ -547,12 +472,6 @@ theorem iso_of_restrict_iso {ℱ ℱ' : Sheaf K A} (α : ℱ ⟶ ℱ') (i : IsIs
   apply (sheaf_hom_eq _ _).symm
 #align category_theory.cover_dense.iso_of_restrict_iso CategoryTheory.CoverDense.iso_of_restrict_iso
 
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 /-- A fully faithful cover-dense functor preserves compatible families. -/
 theorem compatiblePreserving [Faithful G] : CompatiblePreserving K G :=
   by
@@ -611,12 +530,6 @@ variable (Hd : CoverDense K G) (Hp : CoverPreserving J K G) (Hl : CoverLifting J
 
 include Hd Hp Hl
 
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 /-- Given a functor between small sites that is cover-dense, cover-preserving, and cover-lifting,
 it induces an equivalence of category of sheaves valued in a complete category.
 -/

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

@@ -281,10 +281,7 @@ theorem appHom_restrict {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) (x) :
 Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.CoverDense.Types.appHom_valid_glueₓ'. -/
 @[simp]
 theorem appHom_valid_glue {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) :
-    appHom H α X ≫ ℱ'.val.map f = ℱ.map f ≫ α.app (op Y) :=
-  by
-  ext
-  apply app_hom_restrict
+    appHom H α X ≫ ℱ'.val.map f = ℱ.map f ≫ α.app (op Y) := by ext; apply app_hom_restrict
 #align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.CoverDense.Types.appHom_valid_glue
 
 /- warning: category_theory.cover_dense.types.app_iso -> CategoryTheory.CoverDense.Types.appIso is a dubious translation:
@@ -302,16 +299,8 @@ noncomputable def appIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val 
     where
   Hom := appHom H i.Hom X
   inv := appHom H i.inv X
-  hom_inv_id' := by
-    ext x
-    apply H.ext
-    intro Y f
-    simp
-  inv_hom_id' := by
-    ext x
-    apply H.ext
-    intro Y f
-    simp
+  hom_inv_id' := by ext x; apply H.ext; intro Y f; simp
+  inv_hom_id' := by ext x; apply H.ext; intro Y f; simp
 #align category_theory.cover_dense.types.app_iso CategoryTheory.CoverDense.Types.appIso
 
 #print CategoryTheory.CoverDense.Types.presheafHom /-
@@ -351,12 +340,8 @@ noncomputable def sheafIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val
     where
   Hom := ⟨(presheafIso H i).Hom⟩
   inv := ⟨(presheafIso H i).inv⟩
-  hom_inv_id' := by
-    ext1
-    apply (presheaf_iso H i).hom_inv_id
-  inv_hom_id' := by
-    ext1
-    apply (presheaf_iso H i).inv_hom_id
+  hom_inv_id' := by ext1; apply (presheaf_iso H i).hom_inv_id
+  inv_hom_id' := by ext1; apply (presheaf_iso H i).inv_hom_id
 #align category_theory.cover_dense.types.sheaf_iso CategoryTheory.CoverDense.Types.sheafIso
 -/
 
@@ -470,12 +455,8 @@ noncomputable def sheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G.op
     where
   Hom := ⟨(presheafIso H i).Hom⟩
   inv := ⟨(presheafIso H i).inv⟩
-  hom_inv_id' := by
-    ext1
-    apply (presheaf_iso H i).hom_inv_id
-  inv_hom_id' := by
-    ext1
-    apply (presheaf_iso H i).inv_hom_id
+  hom_inv_id' := by ext1; apply (presheaf_iso H i).hom_inv_id
+  inv_hom_id' := by ext1; apply (presheaf_iso H i).inv_hom_id
 #align category_theory.cover_dense.sheaf_iso CategoryTheory.CoverDense.sheafIso
 -/
 

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

@@ -125,10 +125,7 @@ variable {K}
 variable {A : Type _} [Category A] {G : C ⥤ D} (H : CoverDense K G)
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.ext CategoryTheory.CoverDense.extₓ'. -/
 -- this is not marked with `@[ext]` because `H` can not be inferred from the type
 theorem ext (H : CoverDense K G) (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.obj (op X)}
@@ -140,10 +137,7 @@ theorem ext (H : CoverDense K G) (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.o
 #align category_theory.cover_dense.ext CategoryTheory.CoverDense.ext
 
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 Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.functor_pullback_pushforward_covering CategoryTheory.CoverDense.functorPullback_pushforward_coveringₓ'. -/
 theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X : C}
     (T : K (G.obj X)) : (T.val.functorPullback G).functorPushforward G ∈ K (G.obj X) :=
@@ -183,10 +177,7 @@ def isoOver {ℱ ℱ' : Sheaf K A} (α : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val)
 -/
 
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 Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.sheaf_eq_amalgamation CategoryTheory.CoverDense.sheaf_eq_amalgamationₓ'. -/
 theorem sheaf_eq_amalgamation (ℱ : Sheaf K A) {X : A} {U : D} {T : Sieve U} (hT)
     (x : FamilyOfElements _ T) (hx) (t) (h : x.IsAmalgamation t) :
@@ -256,10 +247,7 @@ noncomputable def appHom (X : D) : ℱ.obj (op X) ⟶ ℱ'.val.obj (op X) := fun
 #align category_theory.cover_dense.types.app_hom CategoryTheory.CoverDense.Types.appHom
 
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 Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.types.pushforward_family_apply CategoryTheory.CoverDense.Types.pushforwardFamily_applyₓ'. -/
 @[simp]
 theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y ⟶ X) :
@@ -275,10 +263,7 @@ theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y 
 #align category_theory.cover_dense.types.pushforward_family_apply CategoryTheory.CoverDense.Types.pushforwardFamily_apply
 
 /- warning: category_theory.cover_dense.types.app_hom_restrict -> CategoryTheory.CoverDense.Types.appHom_restrict is a dubious translation:
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 Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.types.app_hom_restrict CategoryTheory.CoverDense.Types.appHom_restrictₓ'. -/
 @[simp]
 theorem appHom_restrict {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) (x) :
@@ -292,10 +277,7 @@ theorem appHom_restrict {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) (x) :
 #align category_theory.cover_dense.types.app_hom_restrict CategoryTheory.CoverDense.Types.appHom_restrict
 
 /- warning: category_theory.cover_dense.types.app_hom_valid_glue -> CategoryTheory.CoverDense.Types.appHom_valid_glue is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.CoverDense.Types.appHom_valid_glueₓ'. -/
 @[simp]
 theorem appHom_valid_glue {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) :
@@ -385,10 +367,7 @@ open Types
 variable {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A}
 
 /- warning: category_theory.cover_dense.sheaf_coyoneda_hom -> CategoryTheory.CoverDense.sheafCoyonedaHom is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.sheaf_coyoneda_hom CategoryTheory.CoverDense.sheafCoyonedaHomₓ'. -/
 /-- (Implementation). The sheaf map given in `types.sheaf_hom` is natural in terms of `X`. -/
 @[simps]

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

@@ -73,7 +73,7 @@ structure Presieve.CoverByImageStructure (G : C ⥤ D) {V U : D} (f : V ⟶ U) w
 
 restate_axiom presieve.cover_by_image_structure.fac'
 
-attribute [simp, reassoc.1] presieve.cover_by_image_structure.fac
+attribute [simp, reassoc] presieve.cover_by_image_structure.fac
 
 #print CategoryTheory.Presieve.coverByImage /-
 /-- For a functor `G : C ⥤ D`, and an object `U : D`, `presieve.cover_by_image G U` is the presieve

mathlib commit https://github.com/leanprover-community/mathlib/commit/c89fe2d59ae06402c3f55f978016d1ada444f57e

@@ -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.dense_subsite
-! leanprover-community/mathlib commit 1d650c2e131f500f3c17f33b4d19d2ea15987f2c
+! leanprover-community/mathlib commit 2ed2c6310e6f1c5562bdf6bfbda55ebbf6891abe
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.CategoryTheory.Adjunction.FullyFaithful
 /-!
 # Dense subsites
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We define `cover_dense` functors into sites as functors such that there exists a covering sieve
 that factors through images of the functor for each object in `D`.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7ad820c4997738e2f542f8a20f32911f52020e26

@@ -56,6 +56,7 @@ variable (J : GrothendieckTopology C) (K : GrothendieckTopology D)
 
 variable {L : GrothendieckTopology E}
 
+#print CategoryTheory.Presieve.CoverByImageStructure /-
 /-- An auxiliary structure that witnesses the fact that `f` factors through an image object of `G`.
 -/
 @[nolint has_nonempty_instance]
@@ -65,18 +66,22 @@ structure Presieve.CoverByImageStructure (G : C ⥤ D) {V U : D} (f : V ⟶ U) w
   map : G.obj obj ⟶ U
   fac : lift ≫ map = f := by obviously
 #align category_theory.presieve.cover_by_image_structure CategoryTheory.Presieve.CoverByImageStructure
+-/
 
 restate_axiom presieve.cover_by_image_structure.fac'
 
 attribute [simp, reassoc.1] presieve.cover_by_image_structure.fac
 
+#print CategoryTheory.Presieve.coverByImage /-
 /-- For a functor `G : C ⥤ D`, and an object `U : D`, `presieve.cover_by_image G U` is the presieve
 of `U` consisting of those arrows that factor through images of `G`.
 -/
 def Presieve.coverByImage (G : C ⥤ D) (U : D) : Presieve U := fun Y f =>
   Nonempty (Presieve.CoverByImageStructure G f)
 #align category_theory.presieve.cover_by_image CategoryTheory.Presieve.coverByImage
+-/
 
+#print CategoryTheory.Sieve.coverByImage /-
 /-- For a functor `G : C ⥤ D`, and an object `U : D`, `sieve.cover_by_image G U` is the sieve of `U`
 consisting of those arrows that factor through images of `G`.
 -/
@@ -84,12 +89,20 @@ def Sieve.coverByImage (G : C ⥤ D) (U : D) : Sieve U :=
   ⟨Presieve.coverByImage G U, fun X Y f ⟨⟨Z, f₁, f₂, (e : _ = _)⟩⟩ g =>
     ⟨⟨Z, g ≫ f₁, f₂, show (g ≫ f₁) ≫ f₂ = g ≫ f by rw [category.assoc, ← e]⟩⟩⟩
 #align category_theory.sieve.cover_by_image CategoryTheory.Sieve.coverByImage
+-/
 
+/- warning: category_theory.presieve.in_cover_by_image -> CategoryTheory.Presieve.in_coverByImage is a dubious translation:
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+  forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {D : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u3} D] (G : CategoryTheory.Functor.{u2, u4, u1, u3} C _inst_1 D _inst_2) {X : D} {Y : C} (f : Quiver.Hom.{succ u4, u3} D (CategoryTheory.CategoryStruct.toQuiver.{u4, u3} D (CategoryTheory.Category.toCategoryStruct.{u4, u3} D _inst_2)) (CategoryTheory.Functor.obj.{u2, u4, u1, u3} C _inst_1 D _inst_2 G Y) X), CategoryTheory.Presieve.coverByImage.{u1, u2, u3, u4} C _inst_1 D _inst_2 G X (CategoryTheory.Functor.obj.{u2, u4, u1, u3} C _inst_1 D _inst_2 G Y) f
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align category_theory.presieve.in_cover_by_image CategoryTheory.Presieve.in_coverByImageₓ'. -/
 theorem Presieve.in_coverByImage (G : C ⥤ D) {X : D} {Y : C} (f : G.obj Y ⟶ X) :
     Presieve.coverByImage G X f :=
   ⟨⟨Y, 𝟙 _, f, by simp⟩⟩
 #align category_theory.presieve.in_cover_by_image CategoryTheory.Presieve.in_coverByImage
 
+#print CategoryTheory.CoverDense /-
 /-- A functor `G : (C, J) ⥤ (D, K)` is called `cover_dense` if for each object in `D`,
   there exists a covering sieve in `D` that factors through images of `G`.
 
@@ -98,6 +111,7 @@ This definition can be found in https://ncatlab.org/nlab/show/dense+sub-site Def
 structure CoverDense (K : GrothendieckTopology D) (G : C ⥤ D) : Prop where
   is_cover : ∀ U : D, Sieve.coverByImage G U ∈ K U
 #align category_theory.cover_dense CategoryTheory.CoverDense
+-/
 
 open Presieve Opposite
 
@@ -107,6 +121,12 @@ variable {K}
 
 variable {A : Type _} [Category A] {G : C ⥤ D} (H : CoverDense K G)
 
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.ext CategoryTheory.CoverDense.extₓ'. -/
 -- this is not marked with `@[ext]` because `H` can not be inferred from the type
 theorem ext (H : CoverDense K G) (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.obj (op X)}
     (h : ∀ ⦃Y : C⦄ (f : G.obj Y ⟶ X), ℱ.val.map f.op s = ℱ.val.map f.op t) : s = t :=
@@ -116,6 +136,12 @@ theorem ext (H : CoverDense K G) (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.o
   simp [h f₂]
 #align category_theory.cover_dense.ext CategoryTheory.CoverDense.ext
 
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.functor_pullback_pushforward_covering CategoryTheory.CoverDense.functorPullback_pushforward_coveringₓ'. -/
 theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X : C}
     (T : K (G.obj X)) : (T.val.functorPullback G).functorPushforward G ∈ K (G.obj X) :=
   by
@@ -127,6 +153,12 @@ theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X :
   · simp
 #align category_theory.cover_dense.functor_pullback_pushforward_covering CategoryTheory.CoverDense.functorPullback_pushforward_covering
 
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.hom_over CategoryTheory.CoverDense.homOverₓ'. -/
 /-- (Implementation). Given an hom between the pullbacks of two sheaves, we can whisker it with
 `coyoneda` to obtain an hom between the pullbacks of the sheaves of maps from `X`.
 -/
@@ -136,6 +168,7 @@ def homOver {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A} (α : G.op ⋙ ℱ ⟶ G.op
   whiskerRight α (coyoneda.obj (op X))
 #align category_theory.cover_dense.hom_over CategoryTheory.CoverDense.homOver
 
+#print CategoryTheory.CoverDense.isoOver /-
 /-- (Implementation). Given an iso between the pullbacks of two sheaves, we can whisker it with
 `coyoneda` to obtain an iso between the pullbacks of the sheaves of maps from `X`.
 -/
@@ -144,7 +177,14 @@ def isoOver {ℱ ℱ' : Sheaf K A} (α : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val)
     G.op ⋙ (sheafOver ℱ X).val ≅ G.op ⋙ (sheafOver ℱ' X).val :=
   isoWhiskerRight α (coyoneda.obj (op X))
 #align category_theory.cover_dense.iso_over CategoryTheory.CoverDense.isoOver
+-/
 
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.sheaf_eq_amalgamation CategoryTheory.CoverDense.sheaf_eq_amalgamationₓ'. -/
 theorem sheaf_eq_amalgamation (ℱ : Sheaf K A) {X : A} {U : D} {T : Sieve U} (hT)
     (x : FamilyOfElements _ T) (hx) (t) (h : x.IsAmalgamation t) :
     t = (ℱ.cond X T hT).amalgamate x hx :=
@@ -157,6 +197,12 @@ namespace Types
 
 variable {ℱ : Dᵒᵖ ⥤ Type v} {ℱ' : SheafOfTypes.{v} K} (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val)
 
+/- warning: category_theory.cover_dense.types.pushforward_family -> CategoryTheory.CoverDense.Types.pushforwardFamily is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.types.pushforward_family CategoryTheory.CoverDense.Types.pushforwardFamilyₓ'. -/
 /--
 (Implementation). Given a section of `ℱ` on `X`, we can obtain a family of elements valued in `ℱ'`
 that is defined on a cover generated by the images of `G`. -/
@@ -168,6 +214,12 @@ noncomputable def pushforwardFamily {X} (x : ℱ.obj (op X)) :
 
 include H
 
+/- warning: category_theory.cover_dense.types.pushforward_family_compatible -> CategoryTheory.CoverDense.Types.pushforwardFamily_compatible is a dubious translation:
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 /-- (Implementation). The `pushforward_family` defined is compatible. -/
 theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) : (pushforwardFamily α x).Compatible :=
   by
@@ -189,11 +241,23 @@ theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) : (pushforwardFami
 
 omit H
 
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.types.app_hom CategoryTheory.CoverDense.Types.appHomₓ'. -/
 /-- (Implementation). The morphism `ℱ(X) ⟶ ℱ'(X)` given by gluing the `pushforward_family`. -/
 noncomputable def appHom (X : D) : ℱ.obj (op X) ⟶ ℱ'.val.obj (op X) := fun x =>
   (ℱ'.cond _ (H.is_cover X)).amalgamate (pushforwardFamily α x) (pushforwardFamily_compatible H α x)
 #align category_theory.cover_dense.types.app_hom CategoryTheory.CoverDense.Types.appHom
 
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.types.pushforward_family_apply CategoryTheory.CoverDense.Types.pushforwardFamily_applyₓ'. -/
 @[simp]
 theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y ⟶ X) :
     pushforwardFamily α x f (Presieve.in_coverByImage G f) = α.app (op Y) (ℱ.map f.op x) :=
@@ -207,6 +271,12 @@ theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y 
     Quiver.Hom.unop_op, ← op_comp, presieve.cover_by_image_structure.fac]
 #align category_theory.cover_dense.types.pushforward_family_apply CategoryTheory.CoverDense.Types.pushforwardFamily_apply
 
+/- warning: category_theory.cover_dense.types.app_hom_restrict -> CategoryTheory.CoverDense.Types.appHom_restrict is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.types.app_hom_restrict CategoryTheory.CoverDense.Types.appHom_restrictₓ'. -/
 @[simp]
 theorem appHom_restrict {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) (x) :
     ℱ'.val.map f (appHom H α X x) = α.app (op Y) (ℱ.map f x) :=
@@ -218,6 +288,12 @@ theorem appHom_restrict {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) (x) :
   apply pushforward_family_apply
 #align category_theory.cover_dense.types.app_hom_restrict CategoryTheory.CoverDense.Types.appHom_restrict
 
+/- warning: category_theory.cover_dense.types.app_hom_valid_glue -> CategoryTheory.CoverDense.Types.appHom_valid_glue is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.CoverDense.Types.appHom_valid_glueₓ'. -/
 @[simp]
 theorem appHom_valid_glue {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) :
     appHom H α X ≫ ℱ'.val.map f = ℱ.map f ≫ α.app (op Y) :=
@@ -226,6 +302,12 @@ theorem appHom_valid_glue {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) :
   apply app_hom_restrict
 #align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.CoverDense.Types.appHom_valid_glue
 
+/- warning: category_theory.cover_dense.types.app_iso -> CategoryTheory.CoverDense.Types.appIso 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 category_theory.cover_dense.types.app_iso CategoryTheory.CoverDense.Types.appIsoₓ'. -/
 /--
 (Implementation). The maps given in `app_iso` is inverse to each other and gives a `ℱ(X) ≅ ℱ'(X)`.
 -/
@@ -247,6 +329,7 @@ noncomputable def appIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val 
     simp
 #align category_theory.cover_dense.types.app_iso CategoryTheory.CoverDense.Types.appIso
 
+#print CategoryTheory.CoverDense.Types.presheafHom /-
 /-- Given an natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of types, where `G` is full
 and cover-dense, and `ℱ'` is a sheaf, we may obtain a natural transformation between sheaves.
 -/
@@ -261,7 +344,9 @@ noncomputable def presheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ 
     simp only [app_hom_restrict, types_comp_apply, ← functor_to_types.map_comp_apply]
     rw [app_hom_restrict H α (f ≫ f'.op : op (unop X) ⟶ _)]
 #align category_theory.cover_dense.types.presheaf_hom CategoryTheory.CoverDense.Types.presheafHom
+-/
 
+#print CategoryTheory.CoverDense.Types.presheafIso /-
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of types, where `G` is full and
 cover-dense, and `ℱ, ℱ'` are sheaves, we may obtain a natural isomorphism between presheaves.
 -/
@@ -270,7 +355,9 @@ noncomputable def presheafIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.
     ℱ.val ≅ ℱ'.val :=
   NatIso.ofComponents (fun X => appIso H i (unop X)) (presheafHom H i.Hom).naturality
 #align category_theory.cover_dense.types.presheaf_iso CategoryTheory.CoverDense.Types.presheafIso
+-/
 
+#print CategoryTheory.CoverDense.Types.sheafIso /-
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of types, where `G` is full and
 cover-dense, and `ℱ, ℱ'` are sheaves, we may obtain a natural isomorphism between sheaves.
 -/
@@ -286,6 +373,7 @@ noncomputable def sheafIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val
     ext1
     apply (presheaf_iso H i).inv_hom_id
 #align category_theory.cover_dense.types.sheaf_iso CategoryTheory.CoverDense.Types.sheafIso
+-/
 
 end Types
 
@@ -293,6 +381,12 @@ open Types
 
 variable {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A}
 
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.sheaf_coyoneda_hom CategoryTheory.CoverDense.sheafCoyonedaHomₓ'. -/
 /-- (Implementation). The sheaf map given in `types.sheaf_hom` is natural in terms of `X`. -/
 @[simps]
 noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
@@ -322,6 +416,7 @@ noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
 
 include H
 
+#print CategoryTheory.CoverDense.sheafYonedaHom /-
 /--
 (Implementation). `sheaf_coyoneda_hom` but the order of the arguments of the functor are swapped.
 -/
@@ -339,9 +434,11 @@ noncomputable def sheafYonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ
     ext (X x)
     exact congr_fun ((α.app X).naturality i) x
 #align category_theory.cover_dense.sheaf_yoneda_hom CategoryTheory.CoverDense.sheafYonedaHom
+-/
 
 omit H
 
+#print CategoryTheory.CoverDense.sheafHom /-
 /-- Given an natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ'` is a sheaf, we may obtain a natural transformation
 between presheaves.
@@ -351,9 +448,11 @@ noncomputable def sheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ ⟶ 
   { app := fun X => yoneda.preimage (α'.app X)
     naturality' := fun X Y f => yoneda.map_injective (by simpa using α'.naturality f) }
 #align category_theory.cover_dense.sheaf_hom CategoryTheory.CoverDense.sheafHom
+-/
 
 include H
 
+#print CategoryTheory.CoverDense.presheafIso /-
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
 we may obtain a natural isomorphism between presheaves.
@@ -375,9 +474,11 @@ noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G
   haveI : is_iso (sheaf_hom H i.hom) := by apply nat_iso.is_iso_of_is_iso_app
   apply as_iso (sheaf_hom H i.hom)
 #align category_theory.cover_dense.presheaf_iso CategoryTheory.CoverDense.presheafIso
+-/
 
 omit H
 
+#print CategoryTheory.CoverDense.sheafIso /-
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
 we may obtain a natural isomorphism between presheaves.
@@ -394,7 +495,14 @@ noncomputable def sheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G.op
     ext1
     apply (presheaf_iso H i).inv_hom_id
 #align category_theory.cover_dense.sheaf_iso CategoryTheory.CoverDense.sheafIso
+-/
 
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.sheaf_hom_restrict_eq CategoryTheory.CoverDense.sheafHom_restrict_eqₓ'. -/
 /-- The constructed `sheaf_hom α` is equal to `α` when restricted onto `C`.
 -/
 theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : whiskerLeft G.op (sheafHom H α) = α :=
@@ -419,6 +527,12 @@ theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : whisker
   infer_instance
 #align category_theory.cover_dense.sheaf_hom_restrict_eq CategoryTheory.CoverDense.sheafHom_restrict_eq
 
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 /-- If the pullback map is obtained via whiskering,
 then the result `sheaf_hom (whisker_left G.op α)` is equal to `α`.
 -/
@@ -438,6 +552,7 @@ theorem sheafHom_eq (α : ℱ ⟶ ℱ'.val) : sheafHom H (whiskerLeft G.op α) =
   simp
 #align category_theory.cover_dense.sheaf_hom_eq CategoryTheory.CoverDense.sheafHom_eq
 
+#print CategoryTheory.CoverDense.restrictHomEquivHom /-
 /-- A full and cover-dense functor `G` induces an equivalence between morphisms into a sheaf and
 morphisms over the restrictions via `G`.
 -/
@@ -448,9 +563,16 @@ noncomputable def restrictHomEquivHom : (G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) ≃
   left_inv := sheafHom_restrict_eq H
   right_inv := sheafHom_eq H
 #align category_theory.cover_dense.restrict_hom_equiv_hom CategoryTheory.CoverDense.restrictHomEquivHom
+-/
 
 include H
 
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+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.iso_of_restrict_iso CategoryTheory.CoverDense.iso_of_restrict_isoₓ'. -/
 /-- Given a full and cover-dense functor `G` and a natural transformation of sheaves `α : ℱ ⟶ ℱ'`,
 if the pullback of `α` along `G` is iso, then `α` is also iso.
 -/
@@ -462,6 +584,12 @@ theorem iso_of_restrict_iso {ℱ ℱ' : Sheaf K A} (α : ℱ ⟶ ℱ') (i : IsIs
   apply (sheaf_hom_eq _ _).symm
 #align category_theory.cover_dense.iso_of_restrict_iso CategoryTheory.CoverDense.iso_of_restrict_iso
 
+/- warning: category_theory.cover_dense.compatible_preserving -> CategoryTheory.CoverDense.compatiblePreserving is a dubious translation:
+lean 3 declaration is
+  forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] {D : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u4, u3} D] {K : CategoryTheory.GrothendieckTopology.{u4, u3} D _inst_2} {G : CategoryTheory.Functor.{u2, u4, u1, u3} C _inst_1 D _inst_2}, (CategoryTheory.CoverDense.{u1, u2, u3, u4} C _inst_1 D _inst_2 K G) -> (forall [_inst_5 : CategoryTheory.Full.{u2, u4, u1, u3} C _inst_1 D _inst_2 G] [_inst_6 : CategoryTheory.Faithful.{u2, u4, u1, u3} C _inst_1 D _inst_2 G], CategoryTheory.CompatiblePreserving.{u5, u2, u4, u1, u3} C _inst_1 D _inst_2 K G)
+but is expected to have type
+  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u5, u3} C] {D : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u4, u2} D] {K : CategoryTheory.GrothendieckTopology.{u4, u2} D _inst_2} {G : CategoryTheory.Functor.{u5, u4, u3, u2} C _inst_1 D _inst_2}, (CategoryTheory.CoverDense.{u3, u5, u2, u4} C _inst_1 D _inst_2 K G) -> (forall [_inst_5 : CategoryTheory.Full.{u5, u4, u3, u2} C _inst_1 D _inst_2 G] [_inst_6 : CategoryTheory.Faithful.{u5, u4, u3, u2} C _inst_1 D _inst_2 G], CategoryTheory.CompatiblePreserving.{u1, u5, u4, u3, u2} C _inst_1 D _inst_2 K G)
+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.compatible_preserving CategoryTheory.CoverDense.compatiblePreservingₓ'. -/
 /-- A fully faithful cover-dense functor preserves compatible families. -/
 theorem compatiblePreserving [Faithful G] : CompatiblePreserving K G :=
   by
@@ -479,13 +607,16 @@ theorem compatiblePreserving [Faithful G] : CompatiblePreserving K G :=
 
 omit H
 
+#print CategoryTheory.CoverDense.Sites.Pullback.full /-
 noncomputable instance Sites.Pullback.full [Faithful G] (Hp : CoverPreserving J K G) :
     Full (Sites.pullback A H.CompatiblePreserving Hp)
     where
   preimage ℱ ℱ' α := ⟨H.sheafHom α.val⟩
   witness' ℱ ℱ' α := Sheaf.Hom.ext _ _ <| H.sheafHom_restrict_eq α.val
 #align category_theory.cover_dense.sites.pullback.full CategoryTheory.CoverDense.Sites.Pullback.full
+-/
 
+#print CategoryTheory.CoverDense.Sites.Pullback.faithful /-
 instance Sites.Pullback.faithful [Faithful G] (Hp : CoverPreserving J K G) :
     Faithful (Sites.pullback A H.CompatiblePreserving Hp)
     where map_injective' := by
@@ -495,6 +626,7 @@ instance Sites.Pullback.faithful [Faithful G] (Hp : CoverPreserving J K G) :
     dsimp at e
     rw [← H.sheaf_hom_eq α.val, ← H.sheaf_hom_eq β.val, e]
 #align category_theory.cover_dense.sites.pullback.faithful CategoryTheory.CoverDense.Sites.Pullback.faithful
+-/
 
 end CoverDense
 
@@ -516,6 +648,12 @@ variable (Hd : CoverDense K G) (Hp : CoverPreserving J K G) (Hl : CoverLifting J
 
 include Hd Hp Hl
 
+/- warning: category_theory.cover_dense.Sheaf_equiv_of_cover_preserving_cover_lifting -> CategoryTheory.CoverDense.sheafEquivOfCoverPreservingCoverLifting is a dubious translation:
+lean 3 declaration is
+  forall {C : Type.{u3}} {D : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u3} D] {G : CategoryTheory.Functor.{u2, u2, u3, u3} C _inst_1 D _inst_2} [_inst_3 : CategoryTheory.Full.{u2, u2, u3, u3} C _inst_1 D _inst_2 G] [_inst_4 : CategoryTheory.Faithful.{u2, u2, u3, u3} C _inst_1 D _inst_2 G] {J : CategoryTheory.GrothendieckTopology.{u2, u3} C _inst_1} {K : CategoryTheory.GrothendieckTopology.{u2, u3} D _inst_2} {A : Type.{u1}} [_inst_5 : CategoryTheory.Category.{max u3 u2, u1} A] [_inst_6 : CategoryTheory.Limits.HasLimits.{max u3 u2, u1} A _inst_5], (CategoryTheory.CoverDense.{u3, u2, u3, u2} C _inst_1 D _inst_2 K G) -> (CategoryTheory.CoverPreserving.{u2, u2, u3, u3} C _inst_1 D _inst_2 J K G) -> (CategoryTheory.CoverLifting.{u3, u2, u3, u2} C _inst_1 D _inst_2 J K G) -> (CategoryTheory.Equivalence.{max u3 u2, max u3 u2, max u3 u1 u3 u2, max u3 u1 u3 u2} (CategoryTheory.Sheaf.{u2, max u3 u2, u3, u1} C _inst_1 J A _inst_5) (CategoryTheory.Sheaf.CategoryTheory.category.{u2, max u3 u2, u3, u1} C _inst_1 J A _inst_5) (CategoryTheory.Sheaf.{u2, max u3 u2, u3, u1} D _inst_2 K A _inst_5) (CategoryTheory.Sheaf.CategoryTheory.category.{u2, max u3 u2, u3, u1} D _inst_2 K A _inst_5))
+but is expected to have type
+  forall {C : Type.{u3}} {D : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u3} D] {G : CategoryTheory.Functor.{u2, u2, u3, u3} C _inst_1 D _inst_2} [_inst_3 : CategoryTheory.Full.{u2, u2, u3, u3} C _inst_1 D _inst_2 G] [_inst_4 : CategoryTheory.Faithful.{u2, u2, u3, u3} C _inst_1 D _inst_2 G] {J : CategoryTheory.GrothendieckTopology.{u2, u3} C _inst_1} {K : CategoryTheory.GrothendieckTopology.{u2, u3} D _inst_2} {A : Type.{u1}} [_inst_5 : CategoryTheory.Category.{max u3 u2, u1} A] [_inst_6 : CategoryTheory.Limits.HasLimits.{max u3 u2, u1} A _inst_5], (CategoryTheory.CoverDense.{u3, u2, u3, u2} C _inst_1 D _inst_2 K G) -> (CategoryTheory.CoverPreserving.{u2, u2, u3, u3} C _inst_1 D _inst_2 J K G) -> (CategoryTheory.CoverLifting.{u3, u2, u3, u2} C _inst_1 D _inst_2 J K G) -> (CategoryTheory.Equivalence.{max u3 u2, max u3 u2, max (max (max u1 u3) u3 u2) u2, max (max (max u1 u3) u3 u2) u2} (CategoryTheory.Sheaf.{u2, max u3 u2, u3, u1} C _inst_1 J A _inst_5) (CategoryTheory.Sheaf.{u2, max u3 u2, u3, u1} D _inst_2 K A _inst_5) (CategoryTheory.Sheaf.instCategorySheaf.{u2, max u3 u2, u3, u1} C _inst_1 J A _inst_5) (CategoryTheory.Sheaf.instCategorySheaf.{u2, max u3 u2, u3, u1} D _inst_2 K A _inst_5))
+Case conversion may be inaccurate. Consider using '#align category_theory.cover_dense.Sheaf_equiv_of_cover_preserving_cover_lifting CategoryTheory.CoverDense.sheafEquivOfCoverPreservingCoverLiftingₓ'. -/
 /-- Given a functor between small sites that is cover-dense, cover-preserving, and cover-lifting,
 it induces an equivalence of category of sheaves valued in a complete category.
 -/

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

@@ -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.dense_subsite
-! leanprover-community/mathlib commit 14b69e9f3c16630440a2cbd46f1ddad0d561dee7
+! leanprover-community/mathlib commit 1d650c2e131f500f3c17f33b4d19d2ea15987f2c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -151,8 +151,6 @@ theorem sheaf_eq_amalgamation (ℱ : Sheaf K A) {X : A} {U : D} {T : Sieve U} (h
   (ℱ.cond X T hT).IsSeparatedFor x t _ h ((ℱ.cond X T hT).IsAmalgamation hx)
 #align category_theory.cover_dense.sheaf_eq_amalgamation CategoryTheory.CoverDense.sheaf_eq_amalgamation
 
-include H
-
 variable [Full G]
 
 namespace Types
@@ -168,9 +166,10 @@ noncomputable def pushforwardFamily {X} (x : ℱ.obj (op X)) :
   ℱ'.val.map hf.some.lift.op <| α.app (op _) (ℱ.map hf.some.map.op x : _)
 #align category_theory.cover_dense.types.pushforward_family CategoryTheory.CoverDense.Types.pushforwardFamily
 
+include H
+
 /-- (Implementation). The `pushforward_family` defined is compatible. -/
-theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) :
-    (pushforwardFamily H α x).Compatible :=
+theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) : (pushforwardFamily α x).Compatible :=
   by
   intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ e
   apply H.ext
@@ -188,15 +187,16 @@ theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) :
   simp [e]
 #align category_theory.cover_dense.types.pushforward_family_compatible CategoryTheory.CoverDense.Types.pushforwardFamily_compatible
 
+omit H
+
 /-- (Implementation). The morphism `ℱ(X) ⟶ ℱ'(X)` given by gluing the `pushforward_family`. -/
 noncomputable def appHom (X : D) : ℱ.obj (op X) ⟶ ℱ'.val.obj (op X) := fun x =>
-  (ℱ'.cond _ (H.is_cover X)).amalgamate (pushforwardFamily H α x)
-    (pushforwardFamily_compatible H α x)
+  (ℱ'.cond _ (H.is_cover X)).amalgamate (pushforwardFamily α x) (pushforwardFamily_compatible H α x)
 #align category_theory.cover_dense.types.app_hom CategoryTheory.CoverDense.Types.appHom
 
 @[simp]
 theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y ⟶ X) :
-    pushforwardFamily H α x f (Presieve.in_coverByImage G f) = α.app (op Y) (ℱ.map f.op x) :=
+    pushforwardFamily α x f (Presieve.in_coverByImage G f) = α.app (op Y) (ℱ.map f.op x) :=
   by
   unfold pushforward_family
   refine' congr_fun _ x
@@ -320,6 +320,8 @@ noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     simp
 #align category_theory.cover_dense.sheaf_coyoneda_hom CategoryTheory.CoverDense.sheafCoyonedaHom
 
+include H
+
 /--
 (Implementation). `sheaf_coyoneda_hom` but the order of the arguments of the functor are swapped.
 -/
@@ -338,6 +340,8 @@ noncomputable def sheafYonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ
     exact congr_fun ((α.app X).naturality i) x
 #align category_theory.cover_dense.sheaf_yoneda_hom CategoryTheory.CoverDense.sheafYonedaHom
 
+omit H
+
 /-- Given an natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ'` is a sheaf, we may obtain a natural transformation
 between presheaves.
@@ -348,6 +352,8 @@ noncomputable def sheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ ⟶ 
     naturality' := fun X Y f => yoneda.map_injective (by simpa using α'.naturality f) }
 #align category_theory.cover_dense.sheaf_hom CategoryTheory.CoverDense.sheafHom
 
+include H
+
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
 we may obtain a natural isomorphism between presheaves.
@@ -370,6 +376,8 @@ noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G
   apply as_iso (sheaf_hom H i.hom)
 #align category_theory.cover_dense.presheaf_iso CategoryTheory.CoverDense.presheafIso
 
+omit H
+
 /-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
 we may obtain a natural isomorphism between presheaves.
@@ -441,6 +449,8 @@ noncomputable def restrictHomEquivHom : (G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) ≃
   right_inv := sheafHom_eq H
 #align category_theory.cover_dense.restrict_hom_equiv_hom CategoryTheory.CoverDense.restrictHomEquivHom
 
+include H
+
 /-- Given a full and cover-dense functor `G` and a natural transformation of sheaves `α : ℱ ⟶ ℱ'`,
 if the pullback of `α` along `G` is iso, then `α` is also iso.
 -/
@@ -467,6 +477,8 @@ theorem compatiblePreserving [Faithful G] : CompatiblePreserving K G :=
   simp [Eq]
 #align category_theory.cover_dense.compatible_preserving CategoryTheory.CoverDense.compatiblePreserving
 
+omit H
+
 noncomputable instance Sites.Pullback.full [Faithful G] (Hp : CoverPreserving J K G) :
     Full (Sites.pullback A H.CompatiblePreserving Hp)
     where

mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee

@@ -63,7 +63,7 @@ structure Presieve.CoverByImageStructure (G : C ⥤ D) {V U : D} (f : V ⟶ U) w
   obj : C
   lift : V ⟶ G.obj obj
   map : G.obj obj ⟶ U
-  fac' : lift ≫ map = f := by obviously
+  fac : lift ≫ map = f := by obviously
 #align category_theory.presieve.cover_by_image_structure CategoryTheory.Presieve.CoverByImageStructure
 
 restate_axiom presieve.cover_by_image_structure.fac'

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

Before this PR, Functor.Full contained the data of the preimage of maps by a full functor F. This PR makes Functor.Full a proposition. This is to prevent any diamond to appear.

The lemma Functor.image_preimage is also renamed Functor.map_preimage.

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

@@ -195,13 +195,13 @@ theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) :
   intro Y f
   simp only [pushforwardFamily, ← FunctorToTypes.map_comp_apply, ← op_comp]
   change (ℱ.map _ ≫ α.app (op _) ≫ ℱ'.val.map _) _ = (ℱ.map _ ≫ α.app (op _) ≫ ℱ'.val.map _) _
-  rw [← G.image_preimage (f ≫ g₁ ≫ _)]
-  rw [← G.image_preimage (f ≫ g₂ ≫ _)]
+  rw [← G.map_preimage (f ≫ g₁ ≫ _)]
+  rw [← G.map_preimage (f ≫ g₂ ≫ _)]
   erw [← α.naturality (G.preimage _).op]
   erw [← α.naturality (G.preimage _).op]
   refine' congr_fun _ x
   simp only [Functor.comp_map, ← Category.assoc, Functor.op_map, Quiver.Hom.unop_op,
-    ← ℱ.map_comp, ← op_comp, G.image_preimage]
+    ← ℱ.map_comp, ← op_comp, G.map_preimage]
   congr 3
   simp [e]
 #align category_theory.cover_dense.types.pushforward_family_compatible CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_compatible
@@ -224,10 +224,10 @@ theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y 
       (α.app (op (Nonempty.some (_ : coverByImage G X f)).1)
         (ℱ.map ((Nonempty.some (_ : coverByImage G X f)).map.op) t))) =
     (fun t => α.app (op Y) (ℱ.map (f.op) t))) x
-  rw [← G.image_preimage (Nonempty.some _ : Presieve.CoverByImageStructure _ _).lift]
+  rw [← G.map_preimage (Nonempty.some _ : Presieve.CoverByImageStructure _ _).lift]
   change ℱ.map _ ≫ α.app (op _) ≫ ℱ'.val.map _ = ℱ.map f.op ≫ α.app (op Y)
   erw [← α.naturality (G.preimage _).op]
-  simp only [← Functor.map_comp, ← Category.assoc, Functor.comp_map, G.image_preimage, G.op_map,
+  simp only [← Functor.map_comp, ← Category.assoc, Functor.comp_map, G.map_preimage, G.op_map,
     Quiver.Hom.unop_op, ← op_comp, Presieve.CoverByImageStructure.fac]
 #align category_theory.cover_dense.types.pushforward_family_apply CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_apply
 
@@ -383,7 +383,7 @@ noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G
       apply isIso_of_reflects_iso _ yoneda
     use (sheafYonedaHom i.inv).app X
     constructor <;> ext x : 2 <;>
-      simp only [sheafHom, NatTrans.comp_app, NatTrans.id_app, Functor.image_preimage]
+      simp only [sheafHom, NatTrans.comp_app, NatTrans.id_app, Functor.map_preimage]
     · exact ((Types.presheafIso (isoOver i (unop x))).app X).hom_inv_id
     · exact ((Types.presheafIso (isoOver i (unop x))).app X).inv_hom_id
     -- Porting note: Lean 4 proof is finished, Lean 3 needed `inferInstance`
@@ -414,8 +414,8 @@ theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
   ext X
   apply yoneda.map_injective
   ext U
-  -- Porting note: didn't need to provide the input to `image_preimage` in Lean 3
-  erw [yoneda.image_preimage ((sheafYonedaHom α).app (G.op.obj X))]
+  -- Porting note: didn't need to provide the input to `map_preimage` in Lean 3
+  erw [yoneda.map_preimage ((sheafYonedaHom α).app (G.op.obj X))]
   symm
   change (show (ℱ'.val ⋙ coyoneda.obj (op (unop U))).obj (op (G.obj (unop X))) from _) = _
   apply sheaf_eq_amalgamation ℱ' (G.is_cover_of_isCoverDense _ _)
@@ -428,7 +428,7 @@ theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
   congr 1
   simp only [Category.assoc]
   congr 1
-  rw [← G.image_preimage hf.some.map]
+  rw [← G.map_preimage hf.some.map]
   symm
   apply α.naturality (G.preimage hf.some.map).op
   -- porting note; Lean 3 needed a random `inferInstance` for cleanup here; not necessary in lean 4
@@ -444,8 +444,8 @@ theorem sheafHom_eq (α : ℱ ⟶ ℱ'.val) : sheafHom (whiskerLeft G.op α) = 
   apply yoneda.map_injective
   -- Porting note: deleted next line as it's not needed in Lean 4
   ext U
-  -- Porting note: Lean 3 didn't need to be told the explicit input to image_preimage
-  erw [yoneda.image_preimage ((sheafYonedaHom (whiskerLeft G.op α)).app X)]
+  -- Porting note: Lean 3 didn't need to be told the explicit input to map_preimage
+  erw [yoneda.map_preimage ((sheafYonedaHom (whiskerLeft G.op α)).app X)]
   symm
   change (show (ℱ'.val ⋙ coyoneda.obj (op (unop U))).obj (op (unop X)) from _) = _
   apply sheaf_eq_amalgamation ℱ' (G.is_cover_of_isCoverDense _ _)
@@ -488,9 +488,9 @@ lemma compatiblePreserving [Faithful G] : CompatiblePreserving K G := by
   apply Functor.IsCoverDense.ext G
   intro W i
   simp only [← FunctorToTypes.map_comp_apply, ← op_comp]
-  rw [← G.image_preimage (i ≫ f₁)]
-  rw [← G.image_preimage (i ≫ f₂)]
-  apply hx
+  rw [← G.map_preimage (i ≫ f₁)]
+  rw [← G.map_preimage (i ≫ f₂)]
+  apply hx (G.preimage (i ≫ f₁)) ((G.preimage (i ≫ f₂))) hg₁ hg₂
   apply G.map_injective
   simp [eq]
 #align category_theory.cover_dense.compatible_preserving CategoryTheory.Functor.IsCoverDense.compatiblePreserving
@@ -498,11 +498,10 @@ lemma compatiblePreserving [Faithful G] : CompatiblePreserving K G := by
 lemma isContinuous [Faithful G] (Hp : CoverPreserving J K G) : G.IsContinuous J K :=
   isContinuous_of_coverPreserving (compatiblePreserving K G) Hp
 
-noncomputable instance fullSheafPushforwardContinuous [G.IsContinuous J K] :
+instance full_sheafPushforwardContinuous [G.IsContinuous J K] :
     Full (G.sheafPushforwardContinuous A J K) where
-  preimage α := ⟨sheafHom α.val⟩
-  witness α := Sheaf.Hom.ext _ _ <| sheafHom_restrict_eq α.val
-#align category_theory.cover_dense.sites.pullback.full CategoryTheory.Functor.IsCoverDense.fullSheafPushforwardContinuous
+  map_surjective α := ⟨⟨sheafHom α.val⟩, Sheaf.Hom.ext _ _ <| sheafHom_restrict_eq α.val⟩
+#align category_theory.cover_dense.sites.pullback.full CategoryTheory.Functor.IsCoverDense.full_sheafPushforwardContinuous
 
 instance faithful_sheafPushforwardContinuous [G.IsContinuous J K] :
     Faithful (G.sheafPushforwardContinuous A J K) where

In all cases, the original proof works now. I presume this is due to simp changes in Lean 4.7, but haven't verified.

@@ -200,12 +200,8 @@ theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) :
   erw [← α.naturality (G.preimage _).op]
   erw [← α.naturality (G.preimage _).op]
   refine' congr_fun _ x
-  -- Porting note: these next 3 tactics (simp, rw, simp) were just one big `simp only` in Lean 3
-  -- but I can't get `simp` to do the `rw` line.
-  simp only [Functor.comp_map, ← Category.assoc, Functor.op_map, Quiver.Hom.unop_op]
-  rw [← ℱ.map_comp, ← ℱ.map_comp] -- `simp only [← ℱ.map_comp]` does nothing, even if I add
-  -- the relevant explicit inputs
-  simp only [← op_comp, G.image_preimage]
+  simp only [Functor.comp_map, ← Category.assoc, Functor.op_map, Quiver.Hom.unop_op,
+    ← ℱ.map_comp, ← op_comp, G.image_preimage]
   congr 3
   simp [e]
 #align category_theory.cover_dense.types.pushforward_family_compatible CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_compatible

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.

@@ -53,7 +53,7 @@ variable {L : GrothendieckTopology E}
 
 /-- An auxiliary structure that witnesses the fact that `f` factors through an image object of `G`.
 -/
--- Porting note: removed `@[nolint has_nonempty_instance]`
+-- Porting note(#5171): removed `@[nolint has_nonempty_instance]`
 structure Presieve.CoverByImageStructure (G : C ⥤ D) {V U : D} (f : V ⟶ U) where
   obj : C
   lift : V ⟶ G.obj obj

@@ -98,6 +98,17 @@ lemma Functor.is_cover_of_isCoverDense (G : C ⥤ D) (K : GrothendieckTopology D
     [G.IsCoverDense K] (U : D) : Sieve.coverByImage G U ∈ K U := by
   apply Functor.IsCoverDense.is_cover
 
+lemma Functor.isCoverDense_of_generate_singleton_functor_π_mem (G : C ⥤ D)
+    (K : GrothendieckTopology D)
+    (h : ∀ B, ∃ (X : C) (f : G.obj X ⟶ B), Sieve.generate (Presieve.singleton f) ∈ K B) :
+    G.IsCoverDense K where
+  is_cover B := by
+    obtain ⟨X, f, h⟩ := h B
+    refine K.superset_covering ?_ h
+    intro Y f ⟨Z, g, _, h, w⟩
+    cases h
+    exact ⟨⟨_, g, _, w⟩⟩
+
 attribute [nolint docBlame] CategoryTheory.Functor.IsCoverDense.is_cover
 
 open Presieve Opposite

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)

@@ -48,9 +48,7 @@ universe w v u
 namespace CategoryTheory
 
 variable {C : Type*} [Category C] {D : Type*} [Category D] {E : Type*} [Category E]
-
 variable (J : GrothendieckTopology C) (K : GrothendieckTopology D)
-
 variable {L : GrothendieckTopology E}
 
 /-- An auxiliary structure that witnesses the fact that `f` factors through an image object of `G`.
@@ -107,7 +105,6 @@ open Presieve Opposite
 namespace Functor.IsCoverDense
 
 variable {K}
-
 variable {A : Type*} [Category A] (G : C ⥤ D) [G.IsCoverDense K]
 
 -- this is not marked with `@[ext]` because `H` can not be inferred from the type
@@ -519,13 +516,9 @@ namespace CategoryTheory.Functor.IsCoverDense
 open CategoryTheory
 
 variable {C D : Type u} [Category.{v} C] [Category.{v} D]
-
 variable (G : C ⥤ D) [Full G] [Faithful G]
-
 variable (J : GrothendieckTopology C) (K : GrothendieckTopology D)
-
 variable {A : Type w} [Category.{max u v} A] [Limits.HasLimits A]
-
 variable [G.IsCoverDense K] [G.IsContinuous J K] [G.IsCocontinuous J K]
 
 instance (Y : Sheaf J A) : IsIso ((G.sheafAdjunctionCocontinuous A J K).counit.app Y) := by

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".

@@ -55,7 +55,7 @@ variable {L : GrothendieckTopology E}
 
 /-- An auxiliary structure that witnesses the fact that `f` factors through an image object of `G`.
 -/
--- porting note: removed `@[nolint has_nonempty_instance]`
+-- Porting note: removed `@[nolint has_nonempty_instance]`
 structure Presieve.CoverByImageStructure (G : C ⥤ D) {V U : D} (f : V ⟶ U) where
   obj : C
   lift : V ⟶ G.obj obj
@@ -164,16 +164,16 @@ variable {ℱ : Dᵒᵖ ⥤ Type v} {ℱ' : SheafOfTypes.{v} K} (α : G.op ⋙ 
 /--
 (Implementation). Given a section of `ℱ` on `X`, we can obtain a family of elements valued in `ℱ'`
 that is defined on a cover generated by the images of `G`. -/
--- porting note: removed `@[simp, nolint unused_arguments]`
+-- Porting note: removed `@[simp, nolint unused_arguments]`
 noncomputable def pushforwardFamily {X} (x : ℱ.obj (op X)) :
     FamilyOfElements ℱ'.val (coverByImage G X) := fun _ _ hf =>
   ℱ'.val.map hf.some.lift.op <| α.app (op _) (ℱ.map hf.some.map.op x : _)
 #align category_theory.cover_dense.types.pushforward_family CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily
 
--- porting note: there are various `include` and `omit`s in this file  (e.g. one is removed here),
+-- Porting note: there are various `include` and `omit`s in this file  (e.g. one is removed here),
 -- none of which are needed in Lean 4.
 
--- porting note: `pushforward_family` was tagged `@[simp]` in Lean 3 so we add the
+-- Porting note: `pushforward_family` was tagged `@[simp]` in Lean 3 so we add the
 -- equation lemma
 @[simp] theorem pushforwardFamily_def {X} (x : ℱ.obj (op X)) :
     pushforwardFamily α x = fun _ _ hf =>
@@ -192,7 +192,7 @@ theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) :
   erw [← α.naturality (G.preimage _).op]
   erw [← α.naturality (G.preimage _).op]
   refine' congr_fun _ x
-  -- porting note: these next 3 tactics (simp, rw, simp) were just one big `simp only` in Lean 3
+  -- Porting note: these next 3 tactics (simp, rw, simp) were just one big `simp only` in Lean 3
   -- but I can't get `simp` to do the `rw` line.
   simp only [Functor.comp_map, ← Category.assoc, Functor.op_map, Quiver.Hom.unop_op]
   rw [← ℱ.map_comp, ← ℱ.map_comp] -- `simp only [← ℱ.map_comp]` does nothing, even if I add
@@ -212,7 +212,7 @@ noncomputable def appHom (X : D) : ℱ.obj (op X) ⟶ ℱ'.val.obj (op X) := fun
 theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y ⟶ X) :
     pushforwardFamily α x f (Presieve.in_coverByImage G f) = α.app (op Y) (ℱ.map f.op x) := by
   unfold pushforwardFamily
-  -- porting note: congr_fun was more powerful in Lean 3; I had to explicitly supply
+  -- Porting note: congr_fun was more powerful in Lean 3; I had to explicitly supply
   -- the type of the first input here even though it's obvious (there is a unique occurrence
   -- of x on each side of the equality)
   refine' congr_fun (_ :
@@ -273,7 +273,7 @@ noncomputable def presheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ 
     apply Functor.IsCoverDense.ext G
     intro Y' f'
     simp only [appHom_restrict, types_comp_apply, ← FunctorToTypes.map_comp_apply]
-    -- porting note: Lean 3 proof continued with a rewrite but we're done here
+    -- Porting note: Lean 3 proof continued with a rewrite but we're done here
 #align category_theory.cover_dense.types.presheaf_hom CategoryTheory.Functor.IsCoverDense.Types.presheafHom
 
 /-- Given a natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of types, where `G` is full
@@ -321,7 +321,7 @@ noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     symm
     apply sheaf_eq_amalgamation
     apply G.is_cover_of_isCoverDense
-    -- porting note: the following line closes a goal which didn't exist before reenableeta
+    -- Porting note: the following line closes a goal which didn't exist before reenableeta
     · exact pushforwardFamily_compatible (homOver α Y.unop) (f.unop ≫ x)
     intro Y' f' hf'
     change unop X ⟶ ℱ.obj (op (unop _)) at x
@@ -372,7 +372,7 @@ noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G
     ℱ.val ≅ ℱ'.val := by
   have : ∀ X : Dᵒᵖ, IsIso ((sheafHom i.hom).app X) := by
     intro X
-    -- porting note: somehow `apply` in Lean 3 is leaving a typeclass goal,
+    -- Porting note: somehow `apply` in Lean 3 is leaving a typeclass goal,
     -- perhaps due to elaboration order. The corresponding `apply` in Lean 4 fails
     -- because the instance can't yet be synthesized. I hence reorder the proof.
     suffices IsIso (yoneda.map ((sheafHom i.hom).app X)) by
@@ -382,7 +382,7 @@ noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G
       simp only [sheafHom, NatTrans.comp_app, NatTrans.id_app, Functor.image_preimage]
     · exact ((Types.presheafIso (isoOver i (unop x))).app X).hom_inv_id
     · exact ((Types.presheafIso (isoOver i (unop x))).app X).inv_hom_id
-    -- porting note: Lean 4 proof is finished, Lean 3 needed `inferInstance`
+    -- Porting note: Lean 4 proof is finished, Lean 3 needed `inferInstance`
   haveI : IsIso (sheafHom i.hom) := by apply NatIso.isIso_of_isIso_app
   apply asIso (sheafHom i.hom)
 #align category_theory.cover_dense.presheaf_iso CategoryTheory.Functor.IsCoverDense.presheafIso
@@ -410,12 +410,12 @@ theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
   ext X
   apply yoneda.map_injective
   ext U
-  -- porting note: didn't need to provide the input to `image_preimage` in Lean 3
+  -- Porting note: didn't need to provide the input to `image_preimage` in Lean 3
   erw [yoneda.image_preimage ((sheafYonedaHom α).app (G.op.obj X))]
   symm
   change (show (ℱ'.val ⋙ coyoneda.obj (op (unop U))).obj (op (G.obj (unop X))) from _) = _
   apply sheaf_eq_amalgamation ℱ' (G.is_cover_of_isCoverDense _ _)
-  -- porting note: next line was not needed in mathlib3
+  -- Porting note: next line was not needed in mathlib3
   · exact (pushforwardFamily_compatible _ _)
   intro Y f hf
   conv_lhs => rw [← hf.some.fac]
@@ -438,14 +438,14 @@ then the result `sheaf_hom (whisker_left G.op α)` is equal to `α`.
 theorem sheafHom_eq (α : ℱ ⟶ ℱ'.val) : sheafHom (whiskerLeft G.op α) = α := by
   ext X
   apply yoneda.map_injective
-  -- porting note: deleted next line as it's not needed in Lean 4
+  -- Porting note: deleted next line as it's not needed in Lean 4
   ext U
-  -- porting note: Lean 3 didn't need to be told the explicit input to image_preimage
+  -- Porting note: Lean 3 didn't need to be told the explicit input to image_preimage
   erw [yoneda.image_preimage ((sheafYonedaHom (whiskerLeft G.op α)).app X)]
   symm
   change (show (ℱ'.val ⋙ coyoneda.obj (op (unop U))).obj (op (unop X)) from _) = _
   apply sheaf_eq_amalgamation ℱ' (G.is_cover_of_isCoverDense _ _)
-  -- porting note: next line was not needed in mathlib3
+  -- Porting note: next line was not needed in mathlib3
   · exact (pushforwardFamily_compatible _ _)
   intro Y f hf
   conv_lhs => rw [← hf.some.fac]

We define a typeclass HasSheafify which says that presheaves on a site with values in some category can be sheafified, i.e. that the inclusion functor from sheaves to presheaves has a left exact left adjoint. We redefine presheafToSheaf as an arbitrary choice of such a left adjoint.

@@ -545,14 +545,7 @@ noncomputable def sheafEquivOfCoverPreservingCoverLifting : Sheaf J A ≌ Sheaf
 set_option linter.uppercaseLean3 false in
 #align category_theory.cover_dense.Sheaf_equiv_of_cover_preserving_cover_lifting CategoryTheory.Functor.IsCoverDense.sheafEquivOfCoverPreservingCoverLifting
 
-variable
-  [ConcreteCategory.{max v u} A]
-  [Limits.PreservesLimits (forget A)]
-  [ReflectsIsomorphisms (forget A)]
-  [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget A)]
-  [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ A]
-  [∀ (X : D), Limits.PreservesColimitsOfShape (K.Cover X)ᵒᵖ (forget A)]
-  [∀ (X : D), Limits.HasColimitsOfShape (K.Cover X)ᵒᵖ A]
+variable [HasWeakSheafify J A] [HasWeakSheafify K A]
 
 /-- The natural isomorphism exhibiting the compatibility of
 `sheafEquivOfCoverPreservingCoverLifting` with sheafification. -/

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>

@@ -11,7 +11,7 @@ import Mathlib.CategoryTheory.Adjunction.FullyFaithful
 /-!
 # Dense subsites
 
-We define `CoverDense` functors into sites as functors such that there exists a covering sieve
+We define `IsCoverDense` functors into sites as functors such that there exists a covering sieve
 that factors through images of the functor for each object in `D`.
 
 We will primarily consider cover-dense functors that are also full, since this notion is in general
@@ -21,15 +21,16 @@ we would need, and some sheafification would be needed for here and there.
 
 ## Main results
 
-- `CategoryTheory.CoverDense.Types.presheafHom`: If `G : C ⥤ (D, K)` is full and cover-dense,
-  then given any presheaf `ℱ` and sheaf `ℱ'` on `D`, and a morphism `α : G ⋙ ℱ ⟶ G ⋙ ℱ'`,
-  we may glue them together to obtain a morphism of presheaves `ℱ ⟶ ℱ'`.
-- `CategoryTheory.CoverDense.sheafIso`: If `ℱ` above is a sheaf and `α` is an iso,
+- `CategoryTheory.Functor.IsCoverDense.Types.presheafHom`: If `G : C ⥤ (D, K)` is full
+  and cover-dense, then given any presheaf `ℱ` and sheaf `ℱ'` on `D`,
+  and a morphism `α : G ⋙ ℱ ⟶ G ⋙ ℱ'`, we may glue them together to obtain
+  a morphism of presheaves `ℱ ⟶ ℱ'`.
+- `CategoryTheory.Functor.IsCoverDense.sheafIso`: If `ℱ` above is a sheaf and `α` is an iso,
   then the result is also an iso.
-- `CategoryTheory.CoverDense.iso_of_restrict_iso`: If `G : C ⥤ (D, K)` is full and cover-dense,
-  then given any sheaves `ℱ, ℱ'` on `D`, and a morphism `α : ℱ ⟶ ℱ'`, then `α` is an iso if
-  `G ⋙ ℱ ⟶ G ⋙ ℱ'` is iso.
-- `CategoryTheory.CoverDense.sheafEquivOfCoverPreservingCoverLifting`:
+- `CategoryTheory.Functor.IsCoverDense.iso_of_restrict_iso`: If `G : C ⥤ (D, K)` is full
+  and cover-dense, then given any sheaves `ℱ, ℱ'` on `D`, and a morphism `α : ℱ ⟶ ℱ'`,
+  then `α` is an iso if `G ⋙ ℱ ⟶ G ⋙ ℱ'` is iso.
+- `CategoryTheory.Functor.IsCoverDense.sheafEquivOfCoverPreservingCoverLifting`:
   If `G : (C, J) ⥤ (D, K)` is fully-faithful, cover-lifting, cover-preserving, and cover-dense,
   then it will induce an equivalence of categories of sheaves valued in a complete category.
 
@@ -86,42 +87,49 @@ theorem Presieve.in_coverByImage (G : C ⥤ D) {X : D} {Y : C} (f : G.obj Y ⟶
   ⟨⟨Y, 𝟙 _, f, by simp⟩⟩
 #align category_theory.presieve.in_cover_by_image CategoryTheory.Presieve.in_coverByImage
 
-/-- A functor `G : (C, J) ⥤ (D, K)` is called `CoverDense` if for each object in `D`,
+/-- A functor `G : (C, J) ⥤ (D, K)` is cover dense if for each object in `D`,
   there exists a covering sieve in `D` that factors through images of `G`.
 
 This definition can be found in https://ncatlab.org/nlab/show/dense+sub-site Definition 2.2.
 -/
-structure CoverDense (K : GrothendieckTopology D) (G : C ⥤ D) : Prop where
+class Functor.IsCoverDense (G : C ⥤ D) (K : GrothendieckTopology D) : Prop where
   is_cover : ∀ U : D, Sieve.coverByImage G U ∈ K U
-#align category_theory.cover_dense CategoryTheory.CoverDense
+#align category_theory.cover_dense CategoryTheory.Functor.IsCoverDense
 
-attribute [nolint docBlame] CategoryTheory.CoverDense.is_cover
+lemma Functor.is_cover_of_isCoverDense (G : C ⥤ D) (K : GrothendieckTopology D)
+    [G.IsCoverDense K] (U : D) : Sieve.coverByImage G U ∈ K U := by
+  apply Functor.IsCoverDense.is_cover
+
+attribute [nolint docBlame] CategoryTheory.Functor.IsCoverDense.is_cover
 
 open Presieve Opposite
 
-namespace CoverDense
+namespace Functor.IsCoverDense
 
 variable {K}
 
-variable {A : Type*} [Category A] {G : C ⥤ D} (H : CoverDense K G)
+variable {A : Type*} [Category A] (G : C ⥤ D) [G.IsCoverDense K]
 
 -- this is not marked with `@[ext]` because `H` can not be inferred from the type
-theorem ext (H : CoverDense K G) (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.obj (op X)}
+theorem ext (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.obj (op X)}
     (h : ∀ ⦃Y : C⦄ (f : G.obj Y ⟶ X), ℱ.val.map f.op s = ℱ.val.map f.op t) : s = t := by
-  apply (ℱ.cond (Sieve.coverByImage G X) (H.is_cover X)).isSeparatedFor.ext
+  apply (ℱ.cond (Sieve.coverByImage G X) (G.is_cover_of_isCoverDense K X)).isSeparatedFor.ext
   rintro Y _ ⟨Z, f₁, f₂, ⟨rfl⟩⟩
   simp [h f₂]
-#align category_theory.cover_dense.ext CategoryTheory.CoverDense.ext
+#align category_theory.cover_dense.ext CategoryTheory.Functor.IsCoverDense.ext
+
+variable {G}
 
-theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X : C}
+theorem functorPullback_pushforward_covering [Full G] {X : C}
     (T : K (G.obj X)) : (T.val.functorPullback G).functorPushforward G ∈ K (G.obj X) := by
-  refine' K.superset_covering _ (K.bind_covering T.property fun Y f _ => H.is_cover Y)
+  refine' K.superset_covering _ (K.bind_covering T.property
+    fun Y f _ => G.is_cover_of_isCoverDense K Y)
   rintro Y _ ⟨Z, _, f, hf, ⟨W, g, f', ⟨rfl⟩⟩, rfl⟩
   use W; use G.preimage (f' ≫ f); use g
   constructor
   · simpa using T.val.downward_closed hf f'
   · simp
-#align category_theory.cover_dense.functor_pullback_pushforward_covering CategoryTheory.CoverDense.functorPullback_pushforward_covering
+#align category_theory.cover_dense.functor_pullback_pushforward_covering CategoryTheory.Functor.IsCoverDense.functorPullback_pushforward_covering
 
 /-- (Implementation). Given a hom between the pullbacks of two sheaves, we can whisker it with
 `coyoneda` to obtain a hom between the pullbacks of the sheaves of maps from `X`.
@@ -130,7 +138,7 @@ theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X :
 def homOver {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A} (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) (X : A) :
     G.op ⋙ ℱ ⋙ coyoneda.obj (op X) ⟶ G.op ⋙ (sheafOver ℱ' X).val :=
   whiskerRight α (coyoneda.obj (op X))
-#align category_theory.cover_dense.hom_over CategoryTheory.CoverDense.homOver
+#align category_theory.cover_dense.hom_over CategoryTheory.Functor.IsCoverDense.homOver
 
 /-- (Implementation). Given an iso between the pullbacks of two sheaves, we can whisker it with
 `coyoneda` to obtain an iso between the pullbacks of the sheaves of maps from `X`.
@@ -139,13 +147,13 @@ def homOver {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A} (α : G.op ⋙ ℱ ⟶ G.op
 def isoOver {ℱ ℱ' : Sheaf K A} (α : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) (X : A) :
     G.op ⋙ (sheafOver ℱ X).val ≅ G.op ⋙ (sheafOver ℱ' X).val :=
   isoWhiskerRight α (coyoneda.obj (op X))
-#align category_theory.cover_dense.iso_over CategoryTheory.CoverDense.isoOver
+#align category_theory.cover_dense.iso_over CategoryTheory.Functor.IsCoverDense.isoOver
 
 theorem sheaf_eq_amalgamation (ℱ : Sheaf K A) {X : A} {U : D} {T : Sieve U} (hT)
     (x : FamilyOfElements _ T) (hx) (t) (h : x.IsAmalgamation t) :
     t = (ℱ.cond X T hT).amalgamate x hx :=
   (ℱ.cond X T hT).isSeparatedFor x t _ h ((ℱ.cond X T hT).isAmalgamation hx)
-#align category_theory.cover_dense.sheaf_eq_amalgamation CategoryTheory.CoverDense.sheaf_eq_amalgamation
+#align category_theory.cover_dense.sheaf_eq_amalgamation CategoryTheory.Functor.IsCoverDense.sheaf_eq_amalgamation
 
 variable [Full G]
 
@@ -160,7 +168,7 @@ that is defined on a cover generated by the images of `G`. -/
 noncomputable def pushforwardFamily {X} (x : ℱ.obj (op X)) :
     FamilyOfElements ℱ'.val (coverByImage G X) := fun _ _ hf =>
   ℱ'.val.map hf.some.lift.op <| α.app (op _) (ℱ.map hf.some.map.op x : _)
-#align category_theory.cover_dense.types.pushforward_family CategoryTheory.CoverDense.Types.pushforwardFamily
+#align category_theory.cover_dense.types.pushforward_family CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily
 
 -- porting note: there are various `include` and `omit`s in this file  (e.g. one is removed here),
 -- none of which are needed in Lean 4.
@@ -175,7 +183,7 @@ noncomputable def pushforwardFamily {X} (x : ℱ.obj (op X)) :
 theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) :
     (pushforwardFamily α x).Compatible := by
   intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ e
-  apply H.ext
+  apply IsCoverDense.ext G
   intro Y f
   simp only [pushforwardFamily, ← FunctorToTypes.map_comp_apply, ← op_comp]
   change (ℱ.map _ ≫ α.app (op _) ≫ ℱ'.val.map _) _ = (ℱ.map _ ≫ α.app (op _) ≫ ℱ'.val.map _) _
@@ -192,13 +200,13 @@ theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) :
   simp only [← op_comp, G.image_preimage]
   congr 3
   simp [e]
-#align category_theory.cover_dense.types.pushforward_family_compatible CategoryTheory.CoverDense.Types.pushforwardFamily_compatible
+#align category_theory.cover_dense.types.pushforward_family_compatible CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_compatible
 
 /-- (Implementation). The morphism `ℱ(X) ⟶ ℱ'(X)` given by gluing the `pushforwardFamily`. -/
 noncomputable def appHom (X : D) : ℱ.obj (op X) ⟶ ℱ'.val.obj (op X) := fun x =>
-  (ℱ'.cond _ (H.is_cover X)).amalgamate (pushforwardFamily α x)
-    (pushforwardFamily_compatible H α x)
-#align category_theory.cover_dense.types.app_hom CategoryTheory.CoverDense.Types.appHom
+  (ℱ'.cond _ (G.is_cover_of_isCoverDense _ X)).amalgamate (pushforwardFamily α x)
+    (pushforwardFamily_compatible α x)
+#align category_theory.cover_dense.types.app_hom CategoryTheory.Functor.IsCoverDense.Types.appHom
 
 @[simp]
 theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y ⟶ X) :
@@ -217,24 +225,22 @@ theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y 
   erw [← α.naturality (G.preimage _).op]
   simp only [← Functor.map_comp, ← Category.assoc, Functor.comp_map, G.image_preimage, G.op_map,
     Quiver.Hom.unop_op, ← op_comp, Presieve.CoverByImageStructure.fac]
-#align category_theory.cover_dense.types.pushforward_family_apply CategoryTheory.CoverDense.Types.pushforwardFamily_apply
+#align category_theory.cover_dense.types.pushforward_family_apply CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_apply
 
 @[simp]
 theorem appHom_restrict {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) (x) :
-    ℱ'.val.map f (appHom H α X x) = α.app (op Y) (ℱ.map f x) := by
-  refine'
-    ((ℱ'.cond _ (H.is_cover X)).valid_glue (pushforwardFamily_compatible H α x) f.unop
-          (Presieve.in_coverByImage G f.unop)).trans
-      _
-  apply pushforwardFamily_apply
-#align category_theory.cover_dense.types.app_hom_restrict CategoryTheory.CoverDense.Types.appHom_restrict
+    ℱ'.val.map f (appHom α X x) = α.app (op Y) (ℱ.map f x) :=
+  ((ℱ'.cond _ (G.is_cover_of_isCoverDense _ X)).valid_glue
+      (pushforwardFamily_compatible α x) f.unop
+          (Presieve.in_coverByImage G f.unop)).trans (pushforwardFamily_apply _ _ _)
+#align category_theory.cover_dense.types.app_hom_restrict CategoryTheory.Functor.IsCoverDense.Types.appHom_restrict
 
 @[simp]
 theorem appHom_valid_glue {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) :
-    appHom H α X ≫ ℱ'.val.map f = ℱ.map f ≫ α.app (op Y) := by
+    appHom α X ≫ ℱ'.val.map f = ℱ.map f ≫ α.app (op Y) := by
   ext
   apply appHom_restrict
-#align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.CoverDense.Types.appHom_valid_glue
+#align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.Functor.IsCoverDense.Types.appHom_valid_glue
 
 /--
 (Implementation). The maps given in `appIso` is inverse to each other and gives a `ℱ(X) ≅ ℱ'(X)`.
@@ -242,33 +248,33 @@ theorem appHom_valid_glue {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) :
 @[simps]
 noncomputable def appIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val)
     (X : D) : ℱ.val.obj (op X) ≅ ℱ'.val.obj (op X) where
-  hom := appHom H i.hom X
-  inv := appHom H i.inv X
+  hom := appHom i.hom X
+  inv := appHom i.inv X
   hom_inv_id := by
     ext x
-    apply H.ext
+    apply Functor.IsCoverDense.ext G
     intro Y f
     simp
   inv_hom_id := by
     ext x
-    apply H.ext
+    apply Functor.IsCoverDense.ext G
     intro Y f
     simp
-#align category_theory.cover_dense.types.app_iso CategoryTheory.CoverDense.Types.appIso
+#align category_theory.cover_dense.types.app_iso CategoryTheory.Functor.IsCoverDense.Types.appIso
 
 /-- Given a natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of types, where `G` is
 full and cover-dense, and `ℱ'` is a sheaf, we may obtain a natural transformation between sheaves.
 -/
 @[simps]
 noncomputable def presheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ ⟶ ℱ'.val where
-  app X := appHom H α (unop X)
+  app X := appHom α (unop X)
   naturality X Y f := by
     ext x
-    apply H.ext ℱ' (unop Y)
+    apply Functor.IsCoverDense.ext G
     intro Y' f'
     simp only [appHom_restrict, types_comp_apply, ← FunctorToTypes.map_comp_apply]
     -- porting note: Lean 3 proof continued with a rewrite but we're done here
-#align category_theory.cover_dense.types.presheaf_hom CategoryTheory.CoverDense.Types.presheafHom
+#align category_theory.cover_dense.types.presheaf_hom CategoryTheory.Functor.IsCoverDense.Types.presheafHom
 
 /-- Given a natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of types, where `G` is full
 and cover-dense, and `ℱ, ℱ'` are sheaves, we may obtain a natural isomorphism between presheaves.
@@ -276,8 +282,8 @@ and cover-dense, and `ℱ, ℱ'` are sheaves, we may obtain a natural isomorphis
 @[simps!]
 noncomputable def presheafIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) :
     ℱ.val ≅ ℱ'.val :=
-  NatIso.ofComponents (fun X => appIso H i (unop X)) @(presheafHom H i.hom).naturality
-#align category_theory.cover_dense.types.presheaf_iso CategoryTheory.CoverDense.Types.presheafIso
+  NatIso.ofComponents (fun X => appIso i (unop X)) @(presheafHom i.hom).naturality
+#align category_theory.cover_dense.types.presheaf_iso CategoryTheory.Functor.IsCoverDense.Types.presheafIso
 
 /-- Given a natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of types, where `G` is full
 and cover-dense, and `ℱ, ℱ'` are sheaves, we may obtain a natural isomorphism between sheaves.
@@ -285,15 +291,15 @@ and cover-dense, and `ℱ, ℱ'` are sheaves, we may obtain a natural isomorphis
 @[simps]
 noncomputable def sheafIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) :
     ℱ ≅ ℱ' where
-  hom := ⟨(presheafIso H i).hom⟩
-  inv := ⟨(presheafIso H i).inv⟩
+  hom := ⟨(presheafIso i).hom⟩
+  inv := ⟨(presheafIso i).inv⟩
   hom_inv_id := by
     ext1
-    apply (presheafIso H i).hom_inv_id
+    apply (presheafIso i).hom_inv_id
   inv_hom_id := by
     ext1
-    apply (presheafIso H i).inv_hom_id
-#align category_theory.cover_dense.types.sheaf_iso CategoryTheory.CoverDense.Types.sheafIso
+    apply (presheafIso i).inv_hom_id
+#align category_theory.cover_dense.types.sheaf_iso CategoryTheory.Functor.IsCoverDense.Types.sheafIso
 
 end Types
 
@@ -306,17 +312,17 @@ variable {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A}
 noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     coyoneda ⋙ (whiskeringLeft Dᵒᵖ A (Type _)).obj ℱ ⟶
       coyoneda ⋙ (whiskeringLeft Dᵒᵖ A (Type _)).obj ℱ'.val where
-  app X := presheafHom H (homOver α (unop X))
+  app X := presheafHom (homOver α (unop X))
   naturality X Y f := by
     ext U x
     change
-      appHom H (homOver α (unop Y)) (unop U) (f.unop ≫ x) =
-        f.unop ≫ appHom H (homOver α (unop X)) (unop U) x
+      appHom (homOver α (unop Y)) (unop U) (f.unop ≫ x) =
+        f.unop ≫ appHom (homOver α (unop X)) (unop U) x
     symm
     apply sheaf_eq_amalgamation
-    apply H.is_cover
+    apply G.is_cover_of_isCoverDense
     -- porting note: the following line closes a goal which didn't exist before reenableeta
-    · exact pushforwardFamily_compatible H (homOver α Y.unop) (f.unop ≫ x)
+    · exact pushforwardFamily_compatible (homOver α Y.unop) (f.unop ≫ x)
     intro Y' f' hf'
     change unop X ⟶ ℱ.obj (op (unop _)) at x
     dsimp
@@ -325,16 +331,16 @@ noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     conv_lhs => rw [← hf'.some.fac]
     simp only [← Category.assoc, op_comp, Functor.map_comp]
     congr 1
-    refine' (appHom_restrict H (homOver α (unop X)) hf'.some.map.op x).trans _
+    refine' (appHom_restrict (homOver α (unop X)) hf'.some.map.op x).trans _
     simp
-#align category_theory.cover_dense.sheaf_coyoneda_hom CategoryTheory.CoverDense.sheafCoyonedaHom
+#align category_theory.cover_dense.sheaf_coyoneda_hom CategoryTheory.Functor.IsCoverDense.sheafCoyonedaHom
 
 /--
 (Implementation). `sheafCoyonedaHom` but the order of the arguments of the functor are swapped.
 -/
 noncomputable def sheafYonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     ℱ ⋙ yoneda ⟶ ℱ'.val ⋙ yoneda := by
-  let α := sheafCoyonedaHom H α
+  let α := sheafCoyonedaHom α
   refine'
     { app := _
       naturality := _ }
@@ -345,17 +351,17 @@ noncomputable def sheafYonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
   · intro U V i
     ext X x
     exact congr_fun ((α.app X).naturality i) x
-#align category_theory.cover_dense.sheaf_yoneda_hom CategoryTheory.CoverDense.sheafYonedaHom
+#align category_theory.cover_dense.sheaf_yoneda_hom CategoryTheory.Functor.IsCoverDense.sheafYonedaHom
 
 /-- Given a natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ'` is a sheaf, we may obtain a natural transformation
 between presheaves.
 -/
 noncomputable def sheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ ⟶ ℱ'.val :=
-  let α' := sheafYonedaHom H α
+  let α' := sheafYonedaHom α
   { app := fun X => yoneda.preimage (α'.app X)
     naturality := fun X Y f => yoneda.map_injective (by simpa using α'.naturality f) }
-#align category_theory.cover_dense.sheaf_hom CategoryTheory.CoverDense.sheafHom
+#align category_theory.cover_dense.sheaf_hom CategoryTheory.Functor.IsCoverDense.sheafHom
 
 /-- Given a natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
@@ -364,22 +370,22 @@ we may obtain a natural isomorphism between presheaves.
 @[simps!]
 noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) :
     ℱ.val ≅ ℱ'.val := by
-  have : ∀ X : Dᵒᵖ, IsIso ((sheafHom H i.hom).app X) := by
+  have : ∀ X : Dᵒᵖ, IsIso ((sheafHom i.hom).app X) := by
     intro X
     -- porting note: somehow `apply` in Lean 3 is leaving a typeclass goal,
     -- perhaps due to elaboration order. The corresponding `apply` in Lean 4 fails
     -- because the instance can't yet be synthesized. I hence reorder the proof.
-    suffices IsIso (yoneda.map ((sheafHom H i.hom).app X)) by
+    suffices IsIso (yoneda.map ((sheafHom i.hom).app X)) by
       apply isIso_of_reflects_iso _ yoneda
-    use (sheafYonedaHom H i.inv).app X
+    use (sheafYonedaHom i.inv).app X
     constructor <;> ext x : 2 <;>
       simp only [sheafHom, NatTrans.comp_app, NatTrans.id_app, Functor.image_preimage]
-    · exact ((Types.presheafIso H (isoOver i (unop x))).app X).hom_inv_id
-    · exact ((Types.presheafIso H (isoOver i (unop x))).app X).inv_hom_id
+    · exact ((Types.presheafIso (isoOver i (unop x))).app X).hom_inv_id
+    · exact ((Types.presheafIso (isoOver i (unop x))).app X).inv_hom_id
     -- porting note: Lean 4 proof is finished, Lean 3 needed `inferInstance`
-  haveI : IsIso (sheafHom H i.hom) := by apply NatIso.isIso_of_isIso_app
-  apply asIso (sheafHom H i.hom)
-#align category_theory.cover_dense.presheaf_iso CategoryTheory.CoverDense.presheafIso
+  haveI : IsIso (sheafHom i.hom) := by apply NatIso.isIso_of_isIso_app
+  apply asIso (sheafHom i.hom)
+#align category_theory.cover_dense.presheaf_iso CategoryTheory.Functor.IsCoverDense.presheafIso
 
 /-- Given a natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
@@ -387,30 +393,30 @@ we may obtain a natural isomorphism between presheaves.
 -/
 @[simps]
 noncomputable def sheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val) : ℱ ≅ ℱ' where
-  hom := ⟨(presheafIso H i).hom⟩
-  inv := ⟨(presheafIso H i).inv⟩
+  hom := ⟨(presheafIso i).hom⟩
+  inv := ⟨(presheafIso i).inv⟩
   hom_inv_id := by
     ext1
-    apply (presheafIso H i).hom_inv_id
+    apply (presheafIso i).hom_inv_id
   inv_hom_id := by
     ext1
-    apply (presheafIso H i).inv_hom_id
-#align category_theory.cover_dense.sheaf_iso CategoryTheory.CoverDense.sheafIso
+    apply (presheafIso i).inv_hom_id
+#align category_theory.cover_dense.sheaf_iso CategoryTheory.Functor.IsCoverDense.sheafIso
 
 /-- The constructed `sheafHom α` is equal to `α` when restricted onto `C`.
 -/
 theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
-    whiskerLeft G.op (sheafHom H α) = α := by
+    whiskerLeft G.op (sheafHom α) = α := by
   ext X
   apply yoneda.map_injective
   ext U
   -- porting note: didn't need to provide the input to `image_preimage` in Lean 3
-  erw [yoneda.image_preimage ((H.sheafYonedaHom α).app (G.op.obj X))]
+  erw [yoneda.image_preimage ((sheafYonedaHom α).app (G.op.obj X))]
   symm
   change (show (ℱ'.val ⋙ coyoneda.obj (op (unop U))).obj (op (G.obj (unop X))) from _) = _
-  apply sheaf_eq_amalgamation ℱ' (H.is_cover _)
+  apply sheaf_eq_amalgamation ℱ' (G.is_cover_of_isCoverDense _ _)
   -- porting note: next line was not needed in mathlib3
-  · exact (pushforwardFamily_compatible H _ _)
+  · exact (pushforwardFamily_compatible _ _)
   intro Y f hf
   conv_lhs => rw [← hf.some.fac]
   simp only [pushforwardFamily, Functor.comp_map, yoneda_map_app, coyoneda_obj_map, op_comp,
@@ -422,54 +428,60 @@ theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
   symm
   apply α.naturality (G.preimage hf.some.map).op
   -- porting note; Lean 3 needed a random `inferInstance` for cleanup here; not necessary in lean 4
-#align category_theory.cover_dense.sheaf_hom_restrict_eq CategoryTheory.CoverDense.sheafHom_restrict_eq
+#align category_theory.cover_dense.sheaf_hom_restrict_eq CategoryTheory.Functor.IsCoverDense.sheafHom_restrict_eq
+
+variable (G)
 
 /-- If the pullback map is obtained via whiskering,
 then the result `sheaf_hom (whisker_left G.op α)` is equal to `α`.
 -/
-theorem sheafHom_eq (α : ℱ ⟶ ℱ'.val) : sheafHom H (whiskerLeft G.op α) = α := by
+theorem sheafHom_eq (α : ℱ ⟶ ℱ'.val) : sheafHom (whiskerLeft G.op α) = α := by
   ext X
   apply yoneda.map_injective
   -- porting note: deleted next line as it's not needed in Lean 4
   ext U
   -- porting note: Lean 3 didn't need to be told the explicit input to image_preimage
-  erw [yoneda.image_preimage ((H.sheafYonedaHom (whiskerLeft G.op α)).app X)]
+  erw [yoneda.image_preimage ((sheafYonedaHom (whiskerLeft G.op α)).app X)]
   symm
   change (show (ℱ'.val ⋙ coyoneda.obj (op (unop U))).obj (op (unop X)) from _) = _
-  apply sheaf_eq_amalgamation ℱ' (H.is_cover _)
+  apply sheaf_eq_amalgamation ℱ' (G.is_cover_of_isCoverDense _ _)
   -- porting note: next line was not needed in mathlib3
-  · exact (pushforwardFamily_compatible H _ _)
+  · exact (pushforwardFamily_compatible _ _)
   intro Y f hf
   conv_lhs => rw [← hf.some.fac]
   dsimp
   simp
-#align category_theory.cover_dense.sheaf_hom_eq CategoryTheory.CoverDense.sheafHom_eq
+#align category_theory.cover_dense.sheaf_hom_eq CategoryTheory.Functor.IsCoverDense.sheafHom_eq
+
+variable {G}
 
 /-- A full and cover-dense functor `G` induces an equivalence between morphisms into a sheaf and
 morphisms over the restrictions via `G`.
 -/
 noncomputable def restrictHomEquivHom : (G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) ≃ (ℱ ⟶ ℱ'.val) where
-  toFun := sheafHom H
+  toFun := sheafHom
   invFun := whiskerLeft G.op
-  left_inv := sheafHom_restrict_eq H
-  right_inv := sheafHom_eq H
-#align category_theory.cover_dense.restrict_hom_equiv_hom CategoryTheory.CoverDense.restrictHomEquivHom
+  left_inv := sheafHom_restrict_eq
+  right_inv := sheafHom_eq _
+#align category_theory.cover_dense.restrict_hom_equiv_hom CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom
 
 /-- Given a full and cover-dense functor `G` and a natural transformation of sheaves `α : ℱ ⟶ ℱ'`,
 if the pullback of `α` along `G` is iso, then `α` is also iso.
 -/
 theorem iso_of_restrict_iso {ℱ ℱ' : Sheaf K A} (α : ℱ ⟶ ℱ') (i : IsIso (whiskerLeft G.op α.val)) :
     IsIso α := by
-  convert IsIso.of_iso (sheafIso H (asIso (whiskerLeft G.op α.val))) using 1
+  convert IsIso.of_iso (sheafIso (asIso (whiskerLeft G.op α.val))) using 1
   ext1
-  apply (sheafHom_eq H _).symm
-#align category_theory.cover_dense.iso_of_restrict_iso CategoryTheory.CoverDense.iso_of_restrict_iso
+  apply (sheafHom_eq _ _).symm
+#align category_theory.cover_dense.iso_of_restrict_iso CategoryTheory.Functor.IsCoverDense.iso_of_restrict_iso
+
+variable (G K)
 
 /-- A fully faithful cover-dense functor preserves compatible families. -/
-theorem compatiblePreserving [Faithful G] : CompatiblePreserving K G := by
+lemma compatiblePreserving [Faithful G] : CompatiblePreserving K G := by
   constructor
   intro ℱ Z T x hx Y₁ Y₂ X f₁ f₂ g₁ g₂ hg₁ hg₂ eq
-  apply H.ext
+  apply Functor.IsCoverDense.ext G
   intro W i
   simp only [← FunctorToTypes.map_comp_apply, ← op_comp]
   rw [← G.image_preimage (i ≫ f₁)]
@@ -477,71 +489,61 @@ theorem compatiblePreserving [Faithful G] : CompatiblePreserving K G := by
   apply hx
   apply G.map_injective
   simp [eq]
-#align category_theory.cover_dense.compatible_preserving CategoryTheory.CoverDense.compatiblePreserving
+#align category_theory.cover_dense.compatible_preserving CategoryTheory.Functor.IsCoverDense.compatiblePreserving
+
+lemma isContinuous [Faithful G] (Hp : CoverPreserving J K G) : G.IsContinuous J K :=
+  isContinuous_of_coverPreserving (compatiblePreserving K G) Hp
 
-noncomputable instance Sites.Pullback.full [Faithful G] (Hp : CoverPreserving J K G) :
-    Full (Sites.pullback A H.compatiblePreserving Hp) where
-  preimage α := ⟨H.sheafHom α.val⟩
-  witness α := Sheaf.Hom.ext _ _ <| H.sheafHom_restrict_eq α.val
-#align category_theory.cover_dense.sites.pullback.full CategoryTheory.CoverDense.Sites.Pullback.full
+noncomputable instance fullSheafPushforwardContinuous [G.IsContinuous J K] :
+    Full (G.sheafPushforwardContinuous A J K) where
+  preimage α := ⟨sheafHom α.val⟩
+  witness α := Sheaf.Hom.ext _ _ <| sheafHom_restrict_eq α.val
+#align category_theory.cover_dense.sites.pullback.full CategoryTheory.Functor.IsCoverDense.fullSheafPushforwardContinuous
 
-instance Sites.Pullback.faithful [Faithful G] (Hp : CoverPreserving J K G) :
-    Faithful (Sites.pullback A H.compatiblePreserving Hp) where
+instance faithful_sheafPushforwardContinuous [G.IsContinuous J K] :
+    Faithful (G.sheafPushforwardContinuous A J K) where
   map_injective := by
     intro ℱ ℱ' α β e
     ext1
     apply_fun fun e => e.val at e
-    dsimp at e
-    rw [← H.sheafHom_eq α.val, ← H.sheafHom_eq β.val, e]
-#align category_theory.cover_dense.sites.pullback.faithful CategoryTheory.CoverDense.Sites.Pullback.faithful
+    dsimp [sheafPushforwardContinuous] at e
+    rw [← sheafHom_eq G α.val, ← sheafHom_eq G β.val, e]
+#align category_theory.cover_dense.sites.pullback.faithful CategoryTheory.Functor.IsCoverDense.faithful_sheafPushforwardContinuous
 
-end CoverDense
+end Functor.IsCoverDense
 
 end CategoryTheory
 
-namespace CategoryTheory.CoverDense
+namespace CategoryTheory.Functor.IsCoverDense
 
 open CategoryTheory
 
 variable {C D : Type u} [Category.{v} C] [Category.{v} D]
 
-variable {G : C ⥤ D} [Full G] [Faithful G]
+variable (G : C ⥤ D) [Full G] [Faithful G]
 
-variable {J : GrothendieckTopology C} {K : GrothendieckTopology D}
+variable (J : GrothendieckTopology C) (K : GrothendieckTopology D)
 
 variable {A : Type w} [Category.{max u v} A] [Limits.HasLimits A]
 
-variable (Hd : CoverDense K G) (Hp : CoverPreserving J K G) (Hl : CoverLifting J K G)
+variable [G.IsCoverDense K] [G.IsContinuous J K] [G.IsCocontinuous J K]
+
+instance (Y : Sheaf J A) : IsIso ((G.sheafAdjunctionCocontinuous A J K).counit.app Y) := by
+    let α := G.sheafAdjunctionCocontinuous A J K
+    haveI : IsIso ((sheafToPresheaf J A).map (α.counit.app Y)) :=
+      IsIso.of_iso ((@asIso _ _ _ _ _ (Ran.reflective A G.op)).app Y.val)
+    apply ReflectsIsomorphisms.reflects (sheafToPresheaf J A)
+
+variable (A)
 
 /-- Given a functor between small sites that is cover-dense, cover-preserving, and cover-lifting,
 it induces an equivalence of category of sheaves valued in a complete category.
 -/
 @[simps! functor inverse]
-noncomputable def sheafEquivOfCoverPreservingCoverLifting : Sheaf J A ≌ Sheaf K A := by
-  symm
-  let α := Sites.pullbackCopullbackAdjunction.{w, v, u} A Hp Hl Hd.compatiblePreserving
-  have : ∀ X : Sheaf J A, IsIso (α.counit.app X) := by
-    intro ℱ
-    -- porting note: I don't know how to do `apply_with foo { instances := ff }`
-    -- so I just create the instance first
-    haveI : IsIso ((sheafToPresheaf J A).map (α.counit.app ℱ)) :=
-      IsIso.of_iso ((@asIso _ _ _ _ _ (Ran.reflective A G.op)).app ℱ.val)
-    apply ReflectsIsomorphisms.reflects (sheafToPresheaf J A)
-  -- porting note: a bunch of instances are not synthesized in lean 4 for some reason
-  haveI : IsIso α.counit := NatIso.isIso_of_isIso_app _
-  haveI : Full (Sites.pullback A Hd.compatiblePreserving Hp) :=
-    CoverDense.Sites.Pullback.full J Hd Hp
-  haveI : Faithful (Sites.pullback A Hd.compatiblePreserving Hp) :=
-    CoverDense.Sites.Pullback.faithful J Hd Hp
-  haveI : IsIso α.unit := CategoryTheory.unit_isIso_of_L_fully_faithful α
-  exact
-    { functor := Sites.pullback A Hd.compatiblePreserving Hp
-      inverse := Sites.copullback A Hl
-      unitIso := asIso α.unit
-      counitIso := asIso α.counit
-      functor_unitIso_comp := fun ℱ => by convert α.left_triangle_components }
+noncomputable def sheafEquivOfCoverPreservingCoverLifting : Sheaf J A ≌ Sheaf K A :=
+  (G.sheafAdjunctionCocontinuous A J K).toEquivalence.symm
 set_option linter.uppercaseLean3 false in
-#align category_theory.cover_dense.Sheaf_equiv_of_cover_preserving_cover_lifting CategoryTheory.CoverDense.sheafEquivOfCoverPreservingCoverLifting
+#align category_theory.cover_dense.Sheaf_equiv_of_cover_preserving_cover_lifting CategoryTheory.Functor.IsCoverDense.sheafEquivOfCoverPreservingCoverLifting
 
 variable
   [ConcreteCategory.{max v u} A]
@@ -556,8 +558,8 @@ variable
 `sheafEquivOfCoverPreservingCoverLifting` with sheafification. -/
 noncomputable
 abbrev sheafEquivOfCoverPreservingCoverLiftingSheafificationCompatibility :
-  (whiskeringLeft _ _ A).obj G.op ⋙ presheafToSheaf _ _ ≅
-  presheafToSheaf _ _ ⋙ (sheafEquivOfCoverPreservingCoverLifting Hd Hp Hl).inverse :=
-Sites.pullbackSheafificationCompatibility _ _ Hl _
+    (whiskeringLeft _ _ A).obj G.op ⋙ presheafToSheaf _ _ ≅
+      presheafToSheaf _ _ ⋙ (sheafEquivOfCoverPreservingCoverLifting G J K A).inverse := by
+  apply Functor.pushforwardContinuousSheafificationCompatibility
 
-end CategoryTheory.CoverDense
+end CategoryTheory.Functor.IsCoverDense

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

This has nice performance benefits.

@@ -46,7 +46,7 @@ universe w v u
 
 namespace CategoryTheory
 
-variable {C : Type _} [Category C] {D : Type _} [Category D] {E : Type _} [Category E]
+variable {C : Type*} [Category C] {D : Type*} [Category D] {E : Type*} [Category E]
 
 variable (J : GrothendieckTopology C) (K : GrothendieckTopology D)
 
@@ -103,7 +103,7 @@ namespace CoverDense
 
 variable {K}
 
-variable {A : Type _} [Category A] {G : C ⥤ D} (H : CoverDense K G)
+variable {A : Type*} [Category A] {G : C ⥤ D} (H : CoverDense K G)
 
 -- this is not marked with `@[ext]` because `H` can not be inferred from the type
 theorem ext (H : CoverDense K G) (ℱ : SheafOfTypes K) (X : D) {s t : ℱ.val.obj (op X)}

I'll close https://github.com/leanprover-community/mathlib/pull/14512 once this is merged.

@@ -543,4 +543,21 @@ noncomputable def sheafEquivOfCoverPreservingCoverLifting : Sheaf J A ≌ Sheaf
 set_option linter.uppercaseLean3 false in
 #align category_theory.cover_dense.Sheaf_equiv_of_cover_preserving_cover_lifting CategoryTheory.CoverDense.sheafEquivOfCoverPreservingCoverLifting
 
+variable
+  [ConcreteCategory.{max v u} A]
+  [Limits.PreservesLimits (forget A)]
+  [ReflectsIsomorphisms (forget A)]
+  [∀ (X : C), Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget A)]
+  [∀ (X : C), Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ A]
+  [∀ (X : D), Limits.PreservesColimitsOfShape (K.Cover X)ᵒᵖ (forget A)]
+  [∀ (X : D), Limits.HasColimitsOfShape (K.Cover X)ᵒᵖ A]
+
+/-- The natural isomorphism exhibiting the compatibility of
+`sheafEquivOfCoverPreservingCoverLifting` with sheafification. -/
+noncomputable
+abbrev sheafEquivOfCoverPreservingCoverLiftingSheafificationCompatibility :
+  (whiskeringLeft _ _ A).obj G.op ⋙ presheafToSheaf _ _ ≅
+  presheafToSheaf _ _ ⋙ (sheafEquivOfCoverPreservingCoverLifting Hd Hp Hl).inverse :=
+Sites.pullbackSheafificationCompatibility _ _ Hl _
+
 end CategoryTheory.CoverDense

• 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.dense_subsite
-! leanprover-community/mathlib commit 1d650c2e131f500f3c17f33b4d19d2ea15987f2c
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.CategoryTheory.Sites.Sheaf
 import Mathlib.CategoryTheory.Sites.CoverLifting
 import Mathlib.CategoryTheory.Adjunction.FullyFaithful
+
+#align_import category_theory.sites.dense_subsite from "leanprover-community/mathlib"@"1d650c2e131f500f3c17f33b4d19d2ea15987f2c"
 /-!
 # Dense subsites
 

Co-authored-by: Xavier Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr> Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Pol'tta / Miyahara Kō <pol_tta@outlook.jp> Co-authored-by: Jason Yuen <jason_yuen2007@hotmail.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com> Co-authored-by: Jireh Loreaux <loreaujy@gmail.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com> Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com> Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

@@ -311,7 +311,7 @@ noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
       coyoneda ⋙ (whiskeringLeft Dᵒᵖ A (Type _)).obj ℱ'.val where
   app X := presheafHom H (homOver α (unop X))
   naturality X Y f := by
-    ext (U x)
+    ext U x
     change
       appHom H (homOver α (unop Y)) (unop U) (f.unop ≫ x) =
         f.unop ≫ appHom H (homOver α (unop X)) (unop U) x
@@ -346,7 +346,7 @@ noncomputable def sheafYonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
       { app := fun X => (α.app X).app U
         naturality := fun X Y f => by simpa using congr_app (α.naturality f) U }
   · intro U V i
-    ext (X x)
+    ext X x
     exact congr_fun ((α.app X).naturality i) x
 #align category_theory.cover_dense.sheaf_yoneda_hom CategoryTheory.CoverDense.sheafYonedaHom
 

Part 2 of #5001

@@ -126,8 +126,8 @@ theorem functorPullback_pushforward_covering [Full G] (H : CoverDense K G) {X :
   · simp
 #align category_theory.cover_dense.functor_pullback_pushforward_covering CategoryTheory.CoverDense.functorPullback_pushforward_covering
 
-/-- (Implementation). Given an hom between the pullbacks of two sheaves, we can whisker it with
-`coyoneda` to obtain an hom between the pullbacks of the sheaves of maps from `X`.
+/-- (Implementation). Given a hom between the pullbacks of two sheaves, we can whisker it with
+`coyoneda` to obtain a hom between the pullbacks of the sheaves of maps from `X`.
 -/
 @[simps!]
 def homOver {ℱ : Dᵒᵖ ⥤ A} {ℱ' : Sheaf K A} (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) (X : A) :
@@ -259,7 +259,7 @@ noncomputable def appIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val 
     simp
 #align category_theory.cover_dense.types.app_iso CategoryTheory.CoverDense.Types.appIso
 
-/-- Given an natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of types, where `G` is
+/-- Given a natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of types, where `G` is
 full and cover-dense, and `ℱ'` is a sheaf, we may obtain a natural transformation between sheaves.
 -/
 @[simps]
@@ -273,7 +273,7 @@ noncomputable def presheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ 
     -- porting note: Lean 3 proof continued with a rewrite but we're done here
 #align category_theory.cover_dense.types.presheaf_hom CategoryTheory.CoverDense.Types.presheafHom
 
-/-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of types, where `G` is full
+/-- Given a natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of types, where `G` is full
 and cover-dense, and `ℱ, ℱ'` are sheaves, we may obtain a natural isomorphism between presheaves.
 -/
 @[simps!]
@@ -282,7 +282,7 @@ noncomputable def presheafIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.
   NatIso.ofComponents (fun X => appIso H i (unop X)) @(presheafHom H i.hom).naturality
 #align category_theory.cover_dense.types.presheaf_iso CategoryTheory.CoverDense.Types.presheafIso
 
-/-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of types, where `G` is full
+/-- Given a natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of types, where `G` is full
 and cover-dense, and `ℱ, ℱ'` are sheaves, we may obtain a natural isomorphism between sheaves.
 -/
 @[simps]
@@ -350,7 +350,7 @@ noncomputable def sheafYonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     exact congr_fun ((α.app X).naturality i) x
 #align category_theory.cover_dense.sheaf_yoneda_hom CategoryTheory.CoverDense.sheafYonedaHom
 
-/-- Given an natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of arbitrary category,
+/-- Given a natural transformation `G ⋙ ℱ ⟶ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ'` is a sheaf, we may obtain a natural transformation
 between presheaves.
 -/
@@ -360,7 +360,7 @@ noncomputable def sheafHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) : ℱ ⟶ 
     naturality := fun X Y f => yoneda.map_injective (by simpa using α'.naturality f) }
 #align category_theory.cover_dense.sheaf_hom CategoryTheory.CoverDense.sheafHom
 
-/-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
+/-- Given a natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
 we may obtain a natural isomorphism between presheaves.
 -/
@@ -384,7 +384,7 @@ noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G
   apply asIso (sheafHom H i.hom)
 #align category_theory.cover_dense.presheaf_iso CategoryTheory.CoverDense.presheafIso
 
-/-- Given an natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
+/-- Given a natural isomorphism `G ⋙ ℱ ≅ G ⋙ ℱ'` between presheaves of arbitrary category,
 where `G` is full and cover-dense, and `ℱ', ℱ` are sheaves,
 we may obtain a natural isomorphism between presheaves.
 -/

These are all doc fixes

@@ -371,7 +371,7 @@ noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G
     intro X
     -- porting note: somehow `apply` in Lean 3 is leaving a typeclass goal,
     -- perhaps due to elaboration order. The corresponding `apply` in Lean 4 fails
-    -- because the instance can't yet be synthezised. I hence reorder the proof.
+    -- because the instance can't yet be synthesized. I hence reorder the proof.
     suffices IsIso (yoneda.map ((sheafHom H i.hom).app X)) by
       apply isIso_of_reflects_iso _ yoneda
     use (sheafYonedaHom H i.inv).app X

@@ -14,7 +14,7 @@ import Mathlib.CategoryTheory.Adjunction.FullyFaithful
 /-!
 # Dense subsites
 
-We define `cover_dense` functors into sites as functors such that there exists a covering sieve
+We define `CoverDense` functors into sites as functors such that there exists a covering sieve
 that factors through images of the functor for each object in `D`.
 
 We will primarily consider cover-dense functors that are also full, since this notion is in general
@@ -69,14 +69,14 @@ attribute [nolint docBlame] Presieve.CoverByImageStructure.obj Presieve.CoverByI
 
 attribute [reassoc (attr := simp)] Presieve.CoverByImageStructure.fac
 
-/-- For a functor `G : C ⥤ D`, and an object `U : D`, `presieve.cover_by_image G U` is the presieve
+/-- For a functor `G : C ⥤ D`, and an object `U : D`, `Presieve.coverByImage G U` is the presieve
 of `U` consisting of those arrows that factor through images of `G`.
 -/
 def Presieve.coverByImage (G : C ⥤ D) (U : D) : Presieve U := fun _ f =>
   Nonempty (Presieve.CoverByImageStructure G f)
 #align category_theory.presieve.cover_by_image CategoryTheory.Presieve.coverByImage
 
-/-- For a functor `G : C ⥤ D`, and an object `U : D`, `sieve.cover_by_image G U` is the sieve of `U`
+/-- For a functor `G : C ⥤ D`, and an object `U : D`, `Sieve.coverByImage G U` is the sieve of `U`
 consisting of those arrows that factor through images of `G`.
 -/
 def Sieve.coverByImage (G : C ⥤ D) (U : D) : Sieve U :=
@@ -89,7 +89,7 @@ theorem Presieve.in_coverByImage (G : C ⥤ D) {X : D} {Y : C} (f : G.obj Y ⟶
   ⟨⟨Y, 𝟙 _, f, by simp⟩⟩
 #align category_theory.presieve.in_cover_by_image CategoryTheory.Presieve.in_coverByImage
 
-/-- A functor `G : (C, J) ⥤ (D, K)` is called `cover_dense` if for each object in `D`,
+/-- A functor `G : (C, J) ⥤ (D, K)` is called `CoverDense` if for each object in `D`,
   there exists a covering sieve in `D` that factors through images of `G`.
 
 This definition can be found in https://ncatlab.org/nlab/show/dense+sub-site Definition 2.2.
@@ -174,7 +174,7 @@ noncomputable def pushforwardFamily {X} (x : ℱ.obj (op X)) :
     pushforwardFamily α x = fun _ _ hf =>
   ℱ'.val.map hf.some.lift.op <| α.app (op _) (ℱ.map hf.some.map.op x : _) := rfl
 
-/-- (Implementation). The `pushforward_family` defined is compatible. -/
+/-- (Implementation). The `pushforwardFamily` defined is compatible. -/
 theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) :
     (pushforwardFamily α x).Compatible := by
   intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ e
@@ -197,7 +197,7 @@ theorem pushforwardFamily_compatible {X} (x : ℱ.obj (op X)) :
   simp [e]
 #align category_theory.cover_dense.types.pushforward_family_compatible CategoryTheory.CoverDense.Types.pushforwardFamily_compatible
 
-/-- (Implementation). The morphism `ℱ(X) ⟶ ℱ'(X)` given by gluing the `pushforward_family`. -/
+/-- (Implementation). The morphism `ℱ(X) ⟶ ℱ'(X)` given by gluing the `pushforwardFamily`. -/
 noncomputable def appHom (X : D) : ℱ.obj (op X) ⟶ ℱ'.val.obj (op X) := fun x =>
   (ℱ'.cond _ (H.is_cover X)).amalgamate (pushforwardFamily α x)
     (pushforwardFamily_compatible H α x)
@@ -210,10 +210,11 @@ theorem pushforwardFamily_apply {X} (x : ℱ.obj (op X)) {Y : C} (f : G.obj Y 
   -- porting note: congr_fun was more powerful in Lean 3; I had to explicitly supply
   -- the type of the first input here even though it's obvious (there is a unique occurrence
   -- of x on each side of the equality)
-  refine' congr_fun (_ : (λ t => ℱ'.val.map ((Nonempty.some (_ : coverByImage G X f)).lift.op)
-    (α.app (op (Nonempty.some (_ : coverByImage G X f)).1)
-      (ℱ.map ((Nonempty.some (_ : coverByImage G X f)).map.op) t))) =
-  (λ t => α.app (op Y) (ℱ.map (f.op) t))) x
+  refine' congr_fun (_ :
+    (fun t => ℱ'.val.map ((Nonempty.some (_ : coverByImage G X f)).lift.op)
+      (α.app (op (Nonempty.some (_ : coverByImage G X f)).1)
+        (ℱ.map ((Nonempty.some (_ : coverByImage G X f)).map.op) t))) =
+    (fun t => α.app (op Y) (ℱ.map (f.op) t))) x
   rw [← G.image_preimage (Nonempty.some _ : Presieve.CoverByImageStructure _ _).lift]
   change ℱ.map _ ≫ α.app (op _) ≫ ℱ'.val.map _ = ℱ.map f.op ≫ α.app (op Y)
   erw [← α.naturality (G.preimage _).op]
@@ -239,7 +240,7 @@ theorem appHom_valid_glue {X : D} {Y : C} (f : op X ⟶ op (G.obj Y)) :
 #align category_theory.cover_dense.types.app_hom_valid_glue CategoryTheory.CoverDense.Types.appHom_valid_glue
 
 /--
-(Implementation). The maps given in `app_iso` is inverse to each other and gives a `ℱ(X) ≅ ℱ'(X)`.
+(Implementation). The maps given in `appIso` is inverse to each other and gives a `ℱ(X) ≅ ℱ'(X)`.
 -/
 @[simps]
 noncomputable def appIso {ℱ ℱ' : SheafOfTypes.{v} K} (i : G.op ⋙ ℱ.val ≅ G.op ⋙ ℱ'.val)
@@ -332,7 +333,7 @@ noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
 #align category_theory.cover_dense.sheaf_coyoneda_hom CategoryTheory.CoverDense.sheafCoyonedaHom
 
 /--
-(Implementation). `sheaf_coyoneda_hom` but the order of the arguments of the functor are swapped.
+(Implementation). `sheafCoyonedaHom` but the order of the arguments of the functor are swapped.
 -/
 noncomputable def sheafYonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     ℱ ⋙ yoneda ⟶ ℱ'.val ⋙ yoneda := by
@@ -376,9 +377,9 @@ noncomputable def presheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G
     use (sheafYonedaHom H i.inv).app X
     constructor <;> ext x : 2 <;>
       simp only [sheafHom, NatTrans.comp_app, NatTrans.id_app, Functor.image_preimage]
-    exact ((Types.presheafIso H (isoOver i (unop x))).app X).hom_inv_id
-    exact ((Types.presheafIso H (isoOver i (unop x))).app X).inv_hom_id
-    -- porting note: Lean 4 proof is finished, Lean 3 needed `infer_instance`
+    · exact ((Types.presheafIso H (isoOver i (unop x))).app X).hom_inv_id
+    · exact ((Types.presheafIso H (isoOver i (unop x))).app X).inv_hom_id
+    -- porting note: Lean 4 proof is finished, Lean 3 needed `inferInstance`
   haveI : IsIso (sheafHom H i.hom) := by apply NatIso.isIso_of_isIso_app
   apply asIso (sheafHom H i.hom)
 #align category_theory.cover_dense.presheaf_iso CategoryTheory.CoverDense.presheafIso
@@ -399,7 +400,7 @@ noncomputable def sheafIso {ℱ ℱ' : Sheaf K A} (i : G.op ⋙ ℱ.val ≅ G.op
     apply (presheafIso H i).inv_hom_id
 #align category_theory.cover_dense.sheaf_iso CategoryTheory.CoverDense.sheafIso
 
-/-- The constructed `sheaf_hom α` is equal to `α` when restricted onto `C`.
+/-- The constructed `sheafHom α` is equal to `α` when restricted onto `C`.
 -/
 theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     whiskerLeft G.op (sheafHom H α) = α := by
@@ -412,7 +413,7 @@ theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
   change (show (ℱ'.val ⋙ coyoneda.obj (op (unop U))).obj (op (G.obj (unop X))) from _) = _
   apply sheaf_eq_amalgamation ℱ' (H.is_cover _)
   -- porting note: next line was not needed in mathlib3
-  { exact (pushforwardFamily_compatible H _ _) }
+  · exact (pushforwardFamily_compatible H _ _)
   intro Y f hf
   conv_lhs => rw [← hf.some.fac]
   simp only [pushforwardFamily, Functor.comp_map, yoneda_map_app, coyoneda_obj_map, op_comp,
@@ -423,7 +424,7 @@ theorem sheafHom_restrict_eq (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
   rw [← G.image_preimage hf.some.map]
   symm
   apply α.naturality (G.preimage hf.some.map).op
-  -- porting note; Lean 3 needed a random `infer_instance` for cleanup here; not necessary in lean 4
+  -- porting note; Lean 3 needed a random `inferInstance` for cleanup here; not necessary in lean 4
 #align category_theory.cover_dense.sheaf_hom_restrict_eq CategoryTheory.CoverDense.sheafHom_restrict_eq
 
 /-- If the pullback map is obtained via whiskering,
@@ -440,7 +441,7 @@ theorem sheafHom_eq (α : ℱ ⟶ ℱ'.val) : sheafHom H (whiskerLeft G.op α) =
   change (show (ℱ'.val ⋙ coyoneda.obj (op (unop U))).obj (op (unop X)) from _) = _
   apply sheaf_eq_amalgamation ℱ' (H.is_cover _)
   -- porting note: next line was not needed in mathlib3
-  { exact (pushforwardFamily_compatible H _ _) }
+  · exact (pushforwardFamily_compatible H _ _)
   intro Y f hf
   conv_lhs => rw [← hf.some.fac]
   dsimp
@@ -492,7 +493,7 @@ instance Sites.Pullback.faithful [Faithful G] (Hp : CoverPreserving J K G) :
   map_injective := by
     intro ℱ ℱ' α β e
     ext1
-    apply_fun fun e => e.val  at e
+    apply_fun fun e => e.val at e
     dsimp at e
     rw [← H.sheafHom_eq α.val, ← H.sheafHom_eq β.val, e]
 #align category_theory.cover_dense.sites.pullback.faithful CategoryTheory.CoverDense.Sites.Pullback.faithful

Now that leanprover/lean4#2210 has been merged, this PR:

• removes all the set_option synthInstance.etaExperiment true commands (and some etaExperiment% term elaborators)
• removes many but not quite all set_option maxHeartbeats commands
• makes various other changes required to cope with leanprover/lean4#2210.

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Matthew Ballard <matt@mrb.email>

@@ -317,6 +317,8 @@ noncomputable def sheafCoyonedaHom (α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val) :
     symm
     apply sheaf_eq_amalgamation
     apply H.is_cover
+    -- porting note: the following line closes a goal which didn't exist before reenableeta
+    · exact pushforwardFamily_compatible H (homOver α Y.unop) (f.unop ≫ x)
     intro Y' f' hf'
     change unop X ⟶ ℱ.obj (op (unop _)) at x
     dsimp

Mostly straightforward. Had to fight with simp once and the typeclass inference system a couple of times.