-- Module: Mathlib.CategoryTheory.Sites.Subcanonical
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
inst_1 : J.Subcanonical
ι : Type u_1
X : ι → C
c : Cofan.{u_1, v, u} X
H : Membership.mem (γ := Set (Sieve.{v, u} c.pt)) ((J : (a : C) → Set (Sieve a)) c.pt) (Sieve.ofArrows X c.inj)
inst_2 : ∀ (i : ι), Mono (c.inj i)
hempty : ∀ (Y : C) (a : IsInitial.{v, u} Y), ⊥ ∈ (J : (a : C) → Set (Sieve a)) Y
hdisj :
  ∀ {i j : ι}, i ≠ j → ∀ {Y : C} (a : Y ⟶ X i) (b : Y ⟶ X j), a ≫ c.inj i = b ≫ c.inj j → Nonempty (IsInitial.{v, u} Y)
heq :
  ∀ (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) {Y : C} {i j : ι} (a : Y ⟶ X i) (b : Y ⟶ X j)
    (hab : a ≫ c.inj i = b ≫ c.inj j),
    Eq (α := (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj i).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X i)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          a :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((s.inj j).hom.app (op Y)) :
            (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X j)).obj.obj (op Y))) →
              (fun (X : Type v) ↦ X) (s.pt.obj.obj (op Y)))
          b :
        (fun (X : Type v) ↦ X) (s.pt.obj (P := Presheaf.IsSheaf J).obj (op Y)))
s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)
Y Z : C
refine_1 :
  ∀ (g : Z ⟶ Y),
    (Sieve.ofArrows X c.inj).functor.map (op g) ≫
        (fun (Y : Cᵒᵖ) ↦
            (↾fun (g : (Sieve.ofArrows X c.inj).functor.obj Y) ↦
                (ConcreteCategory.hom (FC := TypeCat.Fun)
                      (NatTrans.app (InducedCategory.Hom.hom (s.inj (Sieve.ofArrows.i ⋯))) Y) :
                    (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X (Sieve.ofArrows.i ⋯))).obj.obj Y)) →
                      (fun (X : Type v) ↦ X) (s.pt.obj.obj Y))
                  (Sieve.ofArrows.h ⋯) :
              (Sieve.ofArrows X c.inj).functor.obj Y ⟶ s.pt.obj (P := Presheaf.IsSheaf J).obj Y))
          (op Z) =
      (fun (Y : Cᵒᵖ) ↦
            (↾fun (g : (Sieve.ofArrows X c.inj).functor.obj Y) ↦
                (ConcreteCategory.hom (FC := TypeCat.Fun)
                      (NatTrans.app (InducedCategory.Hom.hom (s.inj (Sieve.ofArrows.i ⋯))) Y) :
                    (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X (Sieve.ofArrows.i ⋯))).obj.obj Y)) →
                      (fun (X : Type v) ↦ X) (s.pt.obj.obj Y))
                  (Sieve.ofArrows.h ⋯) :
              (Sieve.ofArrows X c.inj).functor.obj Y ⟶ s.pt.obj (P := Presheaf.IsSheaf J).obj Y))
          (op Y) ≫
        s.pt.obj (P := Presheaf.IsSheaf J).map (op g)
⊢ ∀ (f : op Y ⟶ op Z),
    (Sieve.ofArrows X c.inj).functor.map f ≫
        (fun (Y : Cᵒᵖ) ↦
            (↾fun (g : (Sieve.ofArrows X c.inj).functor.obj Y) ↦
                (ConcreteCategory.hom (FC := TypeCat.Fun)
                      (NatTrans.app (InducedCategory.Hom.hom (s.inj (Sieve.ofArrows.i ⋯))) Y) :
                    (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X (Sieve.ofArrows.i ⋯))).obj.obj Y)) →
                      (fun (X : Type v) ↦ X) (s.pt.obj.obj Y))
                  (Sieve.ofArrows.h ⋯) :
              (Sieve.ofArrows X c.inj).functor.obj Y ⟶ s.pt.obj (P := Presheaf.IsSheaf J).obj Y))
          (op Z) =
      (fun (Y : Cᵒᵖ) ↦
            (↾fun (g : (Sieve.ofArrows X c.inj).functor.obj Y) ↦
                (ConcreteCategory.hom (FC := TypeCat.Fun)
                      (NatTrans.app (InducedCategory.Hom.hom (s.inj (Sieve.ofArrows.i ⋯))) Y) :
                    (a : (fun (X : Type v) ↦ X) ((J.yoneda.obj (X (Sieve.ofArrows.i ⋯))).obj.obj Y)) →
                      (fun (X : Type v) ↦ X) (s.pt.obj.obj Y))
                  (Sieve.ofArrows.h ⋯) :
              (Sieve.ofArrows X c.inj).functor.obj Y ⟶ s.pt.obj (P := Presheaf.IsSheaf J).obj Y))
          (op Y) ≫
        s.pt.obj (P := Presheaf.IsSheaf J).map f

Tactic:
⟨(g : Z ⟶ Y)⟩