-- 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
left :
  ∀ (V : C) (iVW : V ⟶ W),
    (G.imageSieve g).arrows iVW →
      ∀ (iVU : V ⟶ U) (h : G.map iVU = G.map iVW ≫ g),
        Eq (α := (fun (X : Type w') ↦ X) ((Y.obj (P := Presheaf.IsSheaf J) ⋙ obj coyoneda (op X)).obj (op V)))
          ((ConcreteCategory.hom (FC := TypeCat.Fun)
                  ((Y.obj (P := Presheaf.IsSheaf J) ⋙ obj coyoneda (op X)).map iVW.op) :
                (a : (fun (X : Type w') ↦ X) ((Y.obj ⋙ coyoneda.obj (op X)).obj (op W))) →
                  (fun (X : Type w') ↦ X) ((Y.obj ⋙ coyoneda.obj (op X)).obj (op V)))
              (f₁ ≫
                NatTrans.app
                  (Limits.Cone.π
                    ((RightExtension.mk (obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J)))
                          (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J)))).coneAt
                      (op (G.obj U))))
                  (StructuredArrow.mk (op g))) :
            (fun (X : Type w') ↦ X) ((Y.obj (P := Presheaf.IsSheaf J) ⋙ obj coyoneda (op X)).obj (op V)))
          ((ConcreteCategory.hom (FC := TypeCat.Fun)
                  ((Y.obj (P := Presheaf.IsSheaf J) ⋙ obj coyoneda (op X)).map iVW.op) :
                (a : (fun (X : Type w') ↦ X) ((Y.obj ⋙ coyoneda.obj (op X)).obj (op W))) →
                  (fun (X : Type w') ↦ X) ((Y.obj ⋙ coyoneda.obj (op X)).obj (op V)))
              (f₂ ≫
                NatTrans.app
                  (Limits.Cone.π
                    ((RightExtension.mk (obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J)))
                          (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J)))).coneAt
                      (op (G.obj U))))
                  (StructuredArrow.mk (op g))) :
            (fun (X : Type w') ↦ X) ((Y.obj (P := Presheaf.IsSheaf J) ⋙ obj coyoneda (op X)).obj (op V)))
⊢ Eq (α :=
    unop (op X) ⟶
      (StructuredArrow.proj (op (G.obj U)) G.op ⋙
            obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj
        (StructuredArrow.mk (op g)))
    (f₁ ≫
      NatTrans.app
        (Limits.Cone.π
          ((RightExtension.mk (obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J)))
                (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J)))).coneAt
            (op (G.obj U))))
        (StructuredArrow.mk (op g)))
    (f₂ ≫
      NatTrans.app
        (Limits.Cone.π
          ((RightExtension.mk (obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J)))
                (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J)))).coneAt
            (op (G.obj U))))
        (StructuredArrow.mk (op g)))

Tactic:
refine (Y.2 X _ (IsDenseSubsite.imageSieve_mem J K G g)).isSeparatedFor.ext
      (fun V iVW ⟨iVU, h⟩ ↦ ?_)

Premises used: CategoryTheory.coyoneda, CategoryTheory.ConcreteCategory.hom, Quiver.Hom.op, CategoryTheory.ObjectProperty.FullSubcategory.property, CategoryTheory.Functor.comp, CategoryTheory.ObjectProperty.FullSubcategory.obj, CategoryTheory.Presieve.IsSeparatedFor.ext, CategoryTheory.Functor.map, CategoryTheory.Presieve.IsSheafFor.isSeparatedFor, _private.0.CategoryTheory.Functor.IsDenseSubsite.isIso_ranCounit_app_of_isDenseSubsite.match_1, Opposite.op, Eq, DFunLike.coe, CategoryTheory.Functor.obj, CategoryTheory.Functor.IsDenseSubsite.imageSieve_mem