-- 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)))
refine_1 :
  (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) →
    (Cocone.pt (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
            (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i))).obj
        (P := Presheaf.IsSheaf J) ⟶
      s.pt.obj (P := Presheaf.IsSheaf J)
refine_2 :
  ∀ (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) (j : ι),
    CategoryStruct.comp (Y :=
        Cocone.pt (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
          (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i)))
        (Cofan.inj (f := fun (i : ι) ↦ J.yoneda.obj (X i))
          (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i)) j)
        ((fun (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) ↦
            ({
                hom :=
                  ⋯.extend
                    {
                      app := 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 ⋯),
                      naturality := ⋯ } } :
              InducedCategory.Hom (D := Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)) (F :=
                ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J))
                (Cocone.pt (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
                  (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i)))
                s.pt))
          s) =
      s.inj j
refine_3 :
  ∀ (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i))
    (m :
      Cocone.pt (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
          (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i)) ⟶
        s.pt)
    (hm :
      ∀ (j : ι),
        Cofan.inj (f := fun (i : ι) ↦ J.yoneda.obj (X i))
              (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i)) j ≫
            m =
          s.inj j),
    m =
      (fun (s : Cofan (C := Sheaf J (Type v)) fun (i : ι) ↦ J.yoneda.obj (X i)) ↦
          ({
              hom :=
                ⋯.extend
                  {
                    app := 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 ⋯),
                    naturality := ⋯ } } :
            InducedCategory.Hom (D := Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)) (F :=
              ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf J))
              (Cocone.pt (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
                (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i)))
              s.pt))
        s
⊢ IsColimit (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ J.yoneda.obj (X i))
    (Cofan.mk (J.yoneda.obj c.pt) fun (i : ι) ↦ J.yoneda.map (c.inj i))

Tactic:
refine Cofan.IsColimit.mk _ (fun s ↦ ⟨?_⟩) (fun s j ↦ ?_) fun s m hm ↦ ?_