-- 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
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
V : C
iUV : V ⟶ r.Z
hiUV : iUV ≫ r.g₁ ≫ l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯ = iUV ≫ r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯
⊢ 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 r.Z))) →
            (fun (X : Type w') ↦ X) ((Y.obj ⋙ coyoneda.obj (op X)).obj (op V)))
        ((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 (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 r.Z))) →
            (fun (X : Type w') ↦ X) ((Y.obj ⋙ coyoneda.obj (op X)).obj (op V)))
        ((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) :
      (fun (X : Type w') ↦ X) ((Y.obj (P := Presheaf.IsSheaf J) ⋙ obj coyoneda (op X)).obj (op V)))

Tactic:
simp [← map_comp, ← op_comp, hiUV]

Premises used: CategoryTheory.op_comp, CategoryTheory.Functor.map_comp