-- 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
c : Limits.Cone (StructuredArrow.proj (op (G.obj U)) G.op ⋙ Y.obj (P := Presheaf.IsSheaf J)) :=
  { pt := X,
    π :=
      {
        app := 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)
            ⋯,
        naturality := ⋯ } }
this :
  CategoryStruct.comp (Y := (obj G.op.ran (Y.obj (P := Presheaf.IsSheaf J))).obj (op (G.obj U)))
      (Limits.IsLimit.lift (G.op.isPointwiseRightKanExtensionRanCounit (Y.obj (P := Presheaf.IsSheaf J)) (op (G.obj U)))
          c :
        c.pt ⟶
          Limits.Cone.pt (J := StructuredArrow (op (G.obj U)) G.op)
            ((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))))
      (NatTrans.app (NatTrans.app G.op.ranCounit (Y.obj (P := Presheaf.IsSheaf J))) (op U)) =
    c.π.app (StructuredArrow.mk (𝟙 (op (G.obj U))))
I :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} U) ↦ S ∈ (J : (a : C) → Set (Sieve a)) U) (G.imageSieve (𝟙 (G.obj U)))
      ⋯)
apply :
  ∀ (V : C) (iUV : V ⟶ I.Y) (hiUV : iUV ≫ l I.Y U U I.f (𝟙 (G.obj U)) ⋯ = iUV ≫ I.f),
    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 iUV.op) :
            (a : (fun (X : Type w') ↦ X) ((Y.obj ⋙ coyoneda.obj (op X)).obj (op I.Y))) →
              (fun (X : Type w') ↦ X) ((Y.obj ⋙ coyoneda.obj (op X)).obj (op V)))
          (⋯.amalgamate
              (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)) ↦
                Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)))
                  ((J : (a : C) → Set (Sieve a)) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)) S)
                (G.imageSieve (V := U) (Quiver.Hom.unop (StructuredArrow.hom (StructuredArrow.mk (𝟙 (op (G.obj U)))))))
                ⋯)
              (fun
                  (I :
                    GrothendieckTopology.Cover.Arrow (X := unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)
                      (Subtype.mk (p :=
                        fun (S : Sieve.{v_1, u_1} (C := C) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)) ↦
                        Membership.mem (γ :=
                          Set (Sieve.{v_1, u_1} (C := C) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)))
                          ((J : (a : C) → Set (Sieve a)) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)) S)
                        (G.imageSieve (V := U)
                          (Quiver.Hom.unop (StructuredArrow.hom (StructuredArrow.mk (𝟙 (op (G.obj U)))))))
                        ⋯)) ↦
                f ≫
                  Y.obj (P := Presheaf.IsSheaf J).map
                    (l I.Y (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2) U I.f
                        (Quiver.Hom.unop (StructuredArrow.hom (StructuredArrow.mk (𝟙 (op (G.obj U)))))) ⋯).op)
              ⋯ ≫
            Y.obj (P := Presheaf.IsSheaf J).map I.f.op) :
        (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 iUV.op) :
            (a : (fun (X : Type w') ↦ X) ((Y.obj ⋙ coyoneda.obj (op X)).obj (op I.Y))) →
              (fun (X : Type w') ↦ X) ((Y.obj ⋙ coyoneda.obj (op X)).obj (op V)))
          (f ≫ Y.obj (P := Presheaf.IsSheaf J).map I.f.op) :
        (fun (X : Type w') ↦ X) ((Y.obj (P := Presheaf.IsSheaf J) ⋙ obj coyoneda (op X)).obj (op V)))
⊢ ⋯.amalgamate
        (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)) ↦
          Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)))
            ((J : (a : C) → Set (Sieve a)) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)) S)
          (G.imageSieve (V := U) (Quiver.Hom.unop (StructuredArrow.hom (StructuredArrow.mk (𝟙 (op (G.obj U))))))) ⋯)
        (fun
            (I :
              GrothendieckTopology.Cover.Arrow (X := unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)) ↦
                  Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)))
                    ((J : (a : C) → Set (Sieve a)) (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2)) S)
                  (G.imageSieve (V := U)
                    (Quiver.Hom.unop (StructuredArrow.hom (StructuredArrow.mk (𝟙 (op (G.obj U)))))))
                  ⋯)) ↦
          f ≫
            Y.obj (P := Presheaf.IsSheaf J).map
              (l I.Y (unop (StructuredArrow.mk (𝟙 (op (G.obj U)))).2) U I.f
                  (Quiver.Hom.unop (StructuredArrow.hom (StructuredArrow.mk (𝟙 (op (G.obj U)))))) ⋯).op)
        ⋯ ≫
      Y.obj (P := Presheaf.IsSheaf J).map I.f.op =
    f ≫ Y.obj (P := Presheaf.IsSheaf J).map I.f.op

Tactic:
apply (Y.2 X _ (IsDenseSubsite.equalizer_mem J K G (l _ _ _ _ _ I.hf)
        I.f (by simp [hl]))).isSeparatedFor.ext fun V iUV (hiUV : _ = _) ↦ ?_

Premises used: CategoryTheory.ObjectProperty.FullSubcategory.property, CategoryTheory.Functor.IsDenseSubsite.equalizer_mem, CategoryTheory.GrothendieckTopology.Cover.Arrow.hf, CategoryTheory.Presieve.IsSeparatedFor.ext, CategoryTheory.Presieve.IsSheafFor.isSeparatedFor, CategoryTheory.GrothendieckTopology.Cover.Arrow.f