-- 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
refine_1 :
  ∀
    (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),
    Eq (α := obj (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) g ⟶ Y.obj (P := Presheaf.IsSheaf J).obj (op r.Z))
      ((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)
          I₁ ≫
        Y.obj (P := Presheaf.IsSheaf J).map r.g₁.op)
      ((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)
          I₂ ≫
        Y.obj (P := Presheaf.IsSheaf J).map r.g₂.op)
⊢ obj (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) g ⟶
    (StructuredArrow.proj (op (G.obj U)) G.op ⋙ Y.obj (P := Presheaf.IsSheaf J)).obj g

Tactic:
refine Y.2.amalgamate ⟨_, IsDenseSubsite.imageSieve_mem J K G g.hom.unop⟩
          (fun I ↦ f ≫ Y.1.map (l _ _ _ _ _ I.hf).op) fun I₁ I₂ r ↦ ?_

Premises used: Quiver.Hom.op, CategoryTheory.ObjectProperty.FullSubcategory.property, Quiver.Hom.unop, CategoryTheory.StructuredArrow.hom, CategoryTheory.GrothendieckTopology.Cover.Arrow.hf, CategoryTheory.ObjectProperty.FullSubcategory.obj, CategoryTheory.Presheaf.IsSheaf.amalgamate, CategoryTheory.Functor.map, Subtype.mk, CategoryTheory.CategoryStruct.comp, CategoryTheory.StructuredArrow.right, CategoryTheory.GrothendieckTopology.Cover.Arrow.f, CategoryTheory.GrothendieckTopology.Cover.Arrow.Y, Opposite.unop, CategoryTheory.Functor.IsDenseSubsite.imageSieve_mem