-- 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
g₁ g₂ : StructuredArrow (op (G.obj U)) G.op
i : g₁ ⟶ g₂
refine_2 :
  ∀ (W₁ : C) (g₁ : unop (G.op.obj (op W₁)) ⟶ unop (op (G.obj U))) (i : StructuredArrow.mk (op g₁) ⟶ g₂),
    Eq (α :=
      obj (C := StructuredArrow (op (G.obj U)) G.op) (obj (const (StructuredArrow (op (G.obj U)) G.op)) X)
          (StructuredArrow.mk (op g₁)) ⟶
        (StructuredArrow.proj (op (G.obj U)) G.op ⋙ Y.obj (P := Presheaf.IsSheaf J)).obj g₂)
      (map (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) i ≫
        (fun (g : StructuredArrow (op (G.obj U)) G.op) ↦
            (⋯.amalgamate
                (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) ⋯)
                (fun
                    (I :
                      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) ⋯)) ↦
                  f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y (unop g.2) U I.f g.hom.unop ⋯).op)
                ⋯ :
              obj (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) g ⟶
                Y.obj (P := Presheaf.IsSheaf J).obj (op (unop g.2))))
          g₂)
      ((fun (g : StructuredArrow (op (G.obj U)) G.op) ↦
            (⋯.amalgamate
                (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) ⋯)
                (fun
                    (I :
                      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) ⋯)) ↦
                  f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y (unop g.2) U I.f g.hom.unop ⋯).op)
                ⋯ :
              obj (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) g ⟶
                Y.obj (P := Presheaf.IsSheaf J).obj (op (unop g.2))))
          (StructuredArrow.mk (op g₁)) ≫
        (StructuredArrow.proj (op (G.obj U)) G.op ⋙ Y.obj (P := Presheaf.IsSheaf J)).map i)
⊢ Eq (α :=
    obj (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) g₁ ⟶
      (StructuredArrow.proj (op (G.obj U)) G.op ⋙ Y.obj (P := Presheaf.IsSheaf J)).obj g₂)
    (map (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) i ≫
      (fun (g : StructuredArrow (op (G.obj U)) G.op) ↦
          (⋯.amalgamate
              (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) ⋯)
              (fun
                  (I :
                    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) ⋯)) ↦
                f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y (unop g.2) U I.f g.hom.unop ⋯).op)
              ⋯ :
            obj (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) g ⟶
              Y.obj (P := Presheaf.IsSheaf J).obj (op (unop g.2))))
        g₂)
    ((fun (g : StructuredArrow (op (G.obj U)) G.op) ↦
          (⋯.amalgamate
              (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) ⋯)
              (fun
                  (I :
                    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) ⋯)) ↦
                f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y (unop g.2) U I.f g.hom.unop ⋯).op)
              ⋯ :
            obj (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) g ⟶
              Y.obj (P := Presheaf.IsSheaf J).obj (op (unop g.2))))
        g₁ ≫
      (StructuredArrow.proj (op (G.obj U)) G.op ⋙ Y.obj (P := Presheaf.IsSheaf J)).map i)

Tactic:
obtain ⟨⟨W₁⟩, ⟨g₁⟩, rfl⟩ := g₁.mk_surjective

Premises used: CategoryTheory.StructuredArrow.mk_surjective, rfl