-- 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 :
  Eq (α :=
    c.pt ⟶
      (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 (𝟙 (op (G.obj U)))))
    (CategoryStruct.comp (Y :=
      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))))
      (Limits.IsLimit.lift (G.op.isPointwiseRightKanExtensionRanCounit (Y.obj (P := Presheaf.IsSheaf J)) (op (G.obj U)))
        c)
      (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 (𝟙 (op (G.obj U))))))
    (c.π.app (StructuredArrow.mk (𝟙 (op (G.obj U)))))
⊢ 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))))

Tactic:
simp only [id_obj, comp_obj, StructuredArrow.proj_obj, StructuredArrow.mk_right,
        RightExtension.coneAt_pt, RightExtension.mk_left, RightExtension.coneAt_π_app,
        const_obj_obj, op_obj, StructuredArrow.mk_hom_eq_self, map_id, whiskeringLeft_obj_obj,
        RightExtension.mk_hom, Category.id_comp] at this

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.Functor.RightExtension.coneAt_pt, CategoryTheory.Functor.RightExtension.coneAt_π_app, CategoryTheory.Functor.RightExtension.mk_hom, CategoryTheory.Functor.RightExtension.mk_left, CategoryTheory.Functor.map_id, CategoryTheory.StructuredArrow.mk_right, CategoryTheory.Functor.id_obj, CategoryTheory.StructuredArrow.mk_hom_eq_self, CategoryTheory.Functor.const_obj_obj, CategoryTheory.StructuredArrow.proj_obj, CategoryTheory.Functor.whiskeringLeft_obj_obj, CategoryTheory.Functor.op_obj, CategoryTheory.Category.id_comp