awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
left :
  Function.Injective (α :=
    (fun (X : Type w') ↦ X)
      (obj
        (obj yoneda
          ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (β :=
    (fun (X : Type w') ↦ X)
      (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
            (op X))) :
      (a :
          (fun (X : Type w') ↦ X)
            ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
        (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
right :
  Function.Surjective (α :=
    (fun (X : Type w') ↦ X)
      (obj
        (obj yoneda
          ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (β :=
    (fun (X : Type w') ↦ X)
      (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
            (op X))) :
      (a :
          (fun (X : Type w') ↦ X)
            ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
        (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
⊢ Function.Bijective (α :=
    (fun (X : Type w') ↦ X)
      (obj
        (obj yoneda
          ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (β :=
    (fun (X : Type w') ↦ X)
      (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
            (op X))) :
      (a :
          (fun (X : Type w') ↦ X)
            ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
        (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))

Tactic:
constructor
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f₁ f₂ :
  obj (obj yoneda ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
    (op X)
e :
  Eq (α :=
    (fun (X : Type w') ↦ X)
      (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app
              (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
              (op X)) :
          (a :
              (fun (X : Type w') ↦ X)
                ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
            (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
        f₁ :
      (fun (X : Type w') ↦ X)
        (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
          (op X)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app
              (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
              (op X)) :
          (a :
              (fun (X : Type w') ↦ X)
                ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
            (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
        f₂ :
      (fun (X : Type w') ↦ X)
        (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
          (op X)))
⊢ f₁ = f₂

Tactic:
(isPointwiseRightKanExtensionRanCounit G.op Y.1 (.op (G.obj U))).hom_ext (fun g ↦ ?_)

Premises used: CategoryTheory.Functor.op, CategoryTheory.Functor.isPointwiseRightKanExtensionRanCounit, CategoryTheory.ObjectProperty.FullSubcategory.obj, CategoryTheory.Limits.IsLimit.hom_ext, Opposite.op, CategoryTheory.Functor.obj
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f₁ f₂ :
  obj (obj yoneda ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
    (op X)
e :
  Eq (α :=
    (fun (X : Type w') ↦ X)
      (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app
              (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
              (op X)) :
          (a :
              (fun (X : Type w') ↦ X)
                ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
            (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
        f₁ :
      (fun (X : Type w') ↦ X)
        (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
          (op X)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app
              (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
              (op X)) :
          (a :
              (fun (X : Type w') ↦ X)
                ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
            (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
        f₂ :
      (fun (X : Type w') ↦ X)
        (obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
          (op X)))
W : C
g : G.obj W ⟶ G.obj U
V : C
iVW : V ⟶ W
h_1 : (G.imageSieve g).arrows iVW
iVU : V ⟶ U
h : G.map iVU = G.map iVW ≫ g
this :
  f₁ ≫
      (G.op ⋙ obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J))).map iVU.op ≫
        NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op V) =
    f₂ ≫
      (G.op ⋙ obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J))).map iVU.op ≫
        NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op V)
⊢ f₁ ≫
      (obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J))).map (op g) ≫
        (G.op ⋙ obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J))).map iVW.op ≫
          NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op V) =
    f₂ ≫
      (obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J))).map (op g) ≫
        (G.op ⋙ obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J))).map iVW.op ≫
          NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op V)

Tactic:
simpa [h] using this
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
⊢ ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    ∃ (l : X ⟶ Z), G.map l = G.map f ≫ g

Tactic:
have (X Y Z) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : G.imageSieve g f) : Exists _ := hf
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
this :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    ∃ (l : X ⟶ Z), G.map l = G.map f ≫ g
right :
  ∀ (l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z)
    (hl :
      ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
        G.map (l X Y Z f g hf) = G.map f ≫ g),
    ∃ (a :
      (fun (X : Type w') ↦ X)
        (obj
          (obj yoneda
            ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
          (op X))),
      (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app
                (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
                (op X)) :
            (a :
                (fun (X : Type w') ↦ X)
                  ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
              (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
          a =
        f
⊢ ∃ (a :
    (fun (X : Type w') ↦ X)
      (obj
        (obj yoneda
          ((obj (G.op.ran ⋙ obj (whiskeringLeft Cᵒᵖ Dᵒᵖ A) G.op) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U)))
        (op X))),
    (ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app
              (map yoneda (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)))
              (op X)) :
          (a :
              (fun (X : Type w') ↦ X)
                ((yoneda.obj (((G.op.ran ⋙ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj Y.obj).obj (op U))).obj (op X))) →
            (fun (X : Type w') ↦ X) ((yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)))
        a =
      f

Tactic:
choose l hl using this
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
g : StructuredArrow (op (G.obj U)) G.op
I₁ I₂ :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
      Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2))) ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
      (G.imageSieve g.hom.unop) ⋯)
r : I₁.Relation I₂
rw : G.map r.g₁ ≫ G.map (l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₁ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₂ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map (l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₃ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
rw₄ : G.map (r.g₁ ≫ I₁.f) ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
rw₅ : G.map (r.g₂ ≫ I₂.f) ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
rw₆ rw₇ : G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
⊢ G.map (r.g₁ ≫ l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)

Tactic:
rw [map_comp, hl, map_comp, hl, ← map_comp_assoc, r.w, map_comp_assoc]

Premises used: CategoryTheory.Functor.map_comp_assoc, CategoryTheory.Functor.map_comp, CategoryTheory.GrothendieckTopology.Cover.Arrow.Relation.w
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
W₁ : C
g₁ : unop (G.op.obj (op W₁)) ⟶ unop (op (G.obj U))
W₂ : C
g₂ : unop (G.op.obj (op W₂)) ⟶ unop (op (G.obj U))
i :
  Quiver.Hom.{v_1, u_1} (V := C)
    (unop
      (StructuredArrow.right.{v_1, v_2, u_1, u_2} (D := Dᵒᵖ) (S := op (G.obj U)) (T := G.op)
        (StructuredArrow.mk (op g₂))))
    (unop
      (StructuredArrow.right.{v_1, v_2, u_1, u_2} (D := Dᵒᵖ) (S := op (G.obj U)) (T := G.op)
        (StructuredArrow.mk (op g₁))))
hi :
  StructuredArrow.hom (StructuredArrow.mk (op g₁)) ≫ G.op.map (op i) = StructuredArrow.hom (StructuredArrow.mk (op g₂))
⊢ (G.op.map (op i)).unop ≫ Quiver.Hom.unop (StructuredArrow.hom (StructuredArrow.mk (op g₁))) = g₂

Tactic:
replace hi := Quiver.Hom.op_inj hi
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
W₁ : C
g₁ : G.obj W₁ ⟶ G.obj U
W₂ : C
i : W₂ ⟶ W₁
hi :
  StructuredArrow.hom (StructuredArrow.mk (op g₁)) ≫ G.op.map (op i) =
    StructuredArrow.hom (StructuredArrow.mk (op (G.map i ≫ g₁)))
refine_2 :
  Eq (α := X ⟶ Y.obj (P := Presheaf.IsSheaf J).obj (op W₂))
    (⋯.amalgamate
      (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
        (G.imageSieve (G.map i ≫ g₁)) ⋯)
      (fun
          (I :
            GrothendieckTopology.Cover.Arrow
              (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
                (G.imageSieve (G.map i ≫ g₁)) ⋯)) ↦
        f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₂ U I.f (G.map i ≫ g₁) ⋯).op)
      ⋯)
    (⋯.amalgamate
        (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁) (G.imageSieve g₁) ⋯)
        (fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁)
                  (G.imageSieve g₁) ⋯)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₁ U I.f g₁ ⋯).op)
        ⋯ ≫
      Y.obj (P := Presheaf.IsSheaf J).map (op i))
⊢ 𝟙 X ≫
      ⋯.amalgamate
        (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
          (G.imageSieve (G.map i ≫ g₁)) ⋯)
        (fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
                  (G.imageSieve (G.map i ≫ g₁)) ⋯)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₂ U I.f (G.map i ≫ g₁) ⋯).op)
        ⋯ =
    ⋯.amalgamate
        (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁) (G.imageSieve g₁) ⋯)
        (fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁)
                  (G.imageSieve g₁) ⋯)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₁ U I.f g₁ ⋯).op)
        ⋯ ≫
      Y.obj (P := Presheaf.IsSheaf J).map (op i)

Tactic:
rw [Category.id_comp]

Premises used: CategoryTheory.Category.id_comp
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
W₁ : C
g₁ : G.obj W₁ ⟶ G.obj U
W₂ : C
i : W₂ ⟶ W₁
hi :
  StructuredArrow.hom (StructuredArrow.mk (op g₁)) ≫ G.op.map (op i) =
    StructuredArrow.hom (StructuredArrow.mk (op (G.map i ≫ g₁)))
I :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
      (G.imageSieve (G.map i ≫ g₁)) ⋯)
refine_2 :
  f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₂ U I.f (G.map i ≫ g₁) ⋯).op =
    ⋯.amalgamate
        (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁) (G.imageSieve g₁) ⋯)
        (fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁)
                  (G.imageSieve g₁) ⋯)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₁ U I.f g₁ ⋯).op)
        ⋯ ≫
      Y.obj (P := Presheaf.IsSheaf J).map (op i ≫ I.f.op)
⊢ Eq (α := X ⟶ Y.obj (P := Presheaf.IsSheaf J).obj (op I.Y))
    (⋯.amalgamate
        (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
          (G.imageSieve (G.map i ≫ g₁)) ⋯)
        (fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
                  (G.imageSieve (G.map i ≫ g₁)) ⋯)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₂ U I.f (G.map i ≫ g₁) ⋯).op)
        ⋯ ≫
      Y.obj (P := Presheaf.IsSheaf J).map I.f.op)
    ((⋯.amalgamate
          (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁) (G.imageSieve g₁) ⋯)
          (fun
              (I :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁)
                    (G.imageSieve g₁) ⋯)) ↦
            f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₁ U I.f g₁ ⋯).op)
          ⋯ ≫
        Y.obj (P := Presheaf.IsSheaf J).map (op i)) ≫
      Y.obj (P := Presheaf.IsSheaf J).map I.f.op)

Tactic:
simp only [Presheaf.IsSheaf.amalgamate_map, Category.assoc, ← Functor.map_comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Presheaf.IsSheaf.amalgamate_map, CategoryTheory.Functor.map_comp
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
W₁ : C
g₁ : G.obj W₁ ⟶ G.obj U
W₂ : C
i : W₂ ⟶ W₁
hi :
  StructuredArrow.hom (StructuredArrow.mk (op g₁)) ≫ G.op.map (op i) =
    StructuredArrow.hom (StructuredArrow.mk (op (G.map i ≫ g₁)))
I :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
      (G.imageSieve (G.map i ≫ g₁)) ⋯)
⊢ G.map (l I.Y W₂ U I.f (G.map i ≫ g₁) ⋯) = G.map (I.f ≫ i) ≫ g₁

Tactic:
simp [hl]