-- 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)))
g : StructuredArrow (op (G.obj U)) G.op
left :
  ∀ (W : C) (g : G.obj W ⟶ G.obj U),
    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)))
⊢ 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
        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))))
        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))))
        g)

Tactic:
obtain ⟨⟨W⟩, ⟨g : G.obj W ⟶ G.obj U⟩, rfl⟩ := g.mk_surjective

Premises used: CategoryTheory.StructuredArrow.mk_surjective, CategoryTheory.Functor.obj, rfl