theorem isVanKampenColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
    {c₁ : Cofan fun i : Fin n ↦ f i.succ} {c₂ : BinaryCofan (f 0) c₁.pt}
    (t₁ : IsVanKampenColimit c₁) (t₂ : IsVanKampenColimit c₂)
    [∀ {Z} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i]
    [HasFiniteCoproducts C] :
    IsVanKampenColimit (extendCofan c₁ c₂) := by
  intro F c α i e hα
  refine ⟨?_, isUniversalColimit_extendCofan f t₁.isUniversal t₂.isUniversal c α i e hα⟩
  intro ⟨Hc⟩ ⟨j⟩
  have t₂' := (@t₂ (pair (F.obj ⟨0⟩) (∐ fun (j : Fin n) ↦ F.obj ⟨j.succ⟩))
    (BinaryCofan.mk (P := c.pt) (c.ι.app _) (Sigma.desc <| fun b ↦ c.ι.app _))
    (mapPair (α.app _) (Sigma.desc <| fun b ↦ α.app _ ≫ c₁.inj _)) i ?_
    (mapPair_equifibered _)).mp ⟨?_⟩
  rotate_left
  · ext ⟨⟨⟩⟩
    · simpa only [pair_obj_left, Functor.const_obj_obj, pair_obj_right, Discrete.functor_obj,
        NatTrans.comp_app, mapPair_left, BinaryCofan.ι_app_left, BinaryCofan.mk_pt,
        BinaryCofan.mk_inl, Functor.const_map_app, extendCofan_pt,
        extendCofan_ι_app, Fin.cases_zero] using congr_app e ⟨0⟩
    · dsimp
      ext j
      simpa only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_ι_app,
        Category.assoc, extendCofan_pt, Functor.const_obj_obj, NatTrans.comp_app, extendCofan_ι_app,
        Fin.cases_succ, Functor.const_map_app] using congr_app e ⟨j.succ⟩
  · let F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
    have : F = Discrete.functor F'
    · apply Functor.hext
      · exact fun i ↦ rfl
      · rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
        simp
    clear_value F'
    subst this
    apply BinaryCofan.IsColimit.mk _ (fun {T} f₁ f₂ ↦ Hc.desc (Cofan.mk T (Fin.cases f₁
      (fun i ↦ Sigma.ι (fun (j : Fin n) ↦ (Discrete.functor F').obj ⟨j.succ⟩) _ ≫ f₂))))
    · intro T f₁ f₂
      simp only [Discrete.functor_obj, pair_obj_left, BinaryCofan.mk_pt, Functor.const_obj_obj,
        BinaryCofan.ι_app_left, BinaryCofan.mk_inl, IsColimit.fac, Cofan.mk_pt, Cofan.mk_ι_app,
        Fin.cases_zero]
    · intro T f₁ f₂
      simp only [Discrete.functor_obj, pair_obj_right, BinaryCofan.mk_pt, Functor.const_obj_obj,
        BinaryCofan.ι_app_right, BinaryCofan.mk_inr]
      ext j
      simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt,
        Cofan.mk_ι_app, IsColimit.fac, Fin.cases_succ]
    · intro T f₁ f₂ f₃ m₁ m₂
      simp at m₁ m₂ ⊢
      refine Hc.uniq (Cofan.mk T (Fin.cases f₁
        (fun i ↦ Sigma.ι (fun (j : Fin n) ↦ (Discrete.functor F').obj ⟨j.succ⟩) _ ≫ f₂))) _ ?_
      intro ⟨j⟩
      simp only [Discrete.functor_obj, Cofan.mk_pt, Functor.const_obj_obj, Cofan.mk_ι_app]
      induction' j using Fin.inductionOn with j _
      · simp only [Fin.cases_zero, m₁]
      · simp only [← m₂, colimit.ι_desc_assoc, Discrete.functor_obj,
          Cofan.mk_pt, Cofan.mk_ι_app, Fin.cases_succ]
  induction' j using Fin.inductionOn with j _
  · exact t₂' ⟨WalkingPair.left⟩
  · have t₁' := (@t₁ (Discrete.functor (fun j ↦ F.obj ⟨j.succ⟩)) (Cofan.mk _ _) (Discrete.natTrans
      <| fun i ↦ α.app _) (Sigma.desc (fun j ↦ α.app _ ≫ c₁.inj _)) ?_
      (NatTrans.equifibered_of_discrete _)).mp ⟨coproductIsCoproduct _⟩ ⟨j⟩
    rotate_left
    · ext ⟨j⟩
      dsimp
      erw [colimit.ι_desc] -- Why?
      

Proof state:
case refine_3.succ.refine_1.w.h.mk
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.318090, u_1} K
D : Type u_2
inst✝² : Category.{?u.318097, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f (Fin.succ i)
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback (BinaryCofan.inr c₂) i
inst✝ : HasFiniteCoproducts C
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
Hc : IsColimit c
t₂' :
  ∀ (j : Discrete WalkingPair),
    IsPullback ((BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b => c.ι.app { as := Fin.succ b })).ι.app j)
      ((mapPair (α.app { as := 0 }) (Sigma.desc fun b => α.app { as := Fin.succ b } ≫ Cofan.inj c₁ b)).app j) i
      (c₂.ι.app j)
j✝ : Fin n
a✝ :
  IsPullback (c.ι.app { as := Fin.castSucc j✝ }) (α.app { as := Fin.castSucc j✝ }) i
    ((extendCofan c₁ c₂).ι.app { as := Fin.castSucc j✝ })
j : Fin n
⊢ α.app { as := Fin.succ j } ≫ c₁.ι.app { as := j } =
    (Cofan.mk c₁.pt fun j =>
            α.app { as := { val := ↑j + 1, isLt := (_ : Nat.succ ↑j < Nat.succ n) } } ≫ Cofan.inj c₁ j).ι.app
      { as := j }

Next tactic:
rfl