-- 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_1 :
  StructuredArrow.hom (StructuredArrow.mk (op g₁)) ≫ G.op.map (op i) = StructuredArrow.hom (StructuredArrow.mk (op g₂))
hi : (G.op.map (op i)).unop ≫ Quiver.Hom.unop (StructuredArrow.hom (StructuredArrow.mk (op g₁))) = g₂
refine_2 :
  𝟙 X ≫
      ⋯.amalgamate
        (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂) (G.imageSieve g₂) ⋯)
        (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 g₂) ⋯)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₂ U I.f g₂ ⋯).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)
⊢ 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 (StructuredArrow.mk (op g₂)))
    (map (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) (StructuredArrow.homMk (op i) hi_1) ≫
      (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 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
        (StructuredArrow.homMk (op i) hi_1))

Tactic:
dsimp at g₁ g₂ i hi ⊢