-- 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 :
  ∀
    (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)) ⋯)),
    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)
⊢ 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))

Tactic:
apply Y.2.hom_ext ⟨_, IsDenseSubsite.imageSieve_mem J K G (G.map i ≫ g₁)⟩

Premises used: CategoryTheory.ObjectProperty.FullSubcategory.property, CategoryTheory.Functor.map, Subtype.mk, CategoryTheory.Presheaf.IsSheaf.hom_ext, CategoryTheory.CategoryStruct.comp, CategoryTheory.Functor.IsDenseSubsite.imageSieve_mem