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 ⟨?_⟩
  

Proof state:
case refine_3
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
j : Fin (n + 1)
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)
⊢ IsPullback (c.ι.app { as := j }) (α.app { as := j }) i ((extendCofan c₁ c₂).ι.app { as := j })
case refine_1
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
j : Fin (n + 1)
⊢ mapPair (α.app { as := 0 }) (Sigma.desc fun b => α.app { as := Fin.succ b } ≫ Cofan.inj c₁ b) ≫ c₂.ι =
    (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b => c.ι.app { as := Fin.succ b })).ι ≫
      (Functor.const (Discrete WalkingPair)).map i
case refine_2
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
j : Fin (n + 1)
⊢ IsColimit (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b => c.ι.app { as := Fin.succ b }))

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