Theorem: CategoryTheory.isVanKampenColimit_extendCofan
File: Mathlib/CategoryTheory/Limits/VanKampen.lean

Proof steps:
state_before: 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst✝ : HasFiniteCoproducts C
⊢ IsVanKampenColimit (extendCofan c₁ c₂)
state_after: 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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
⊢ Nonempty (IsColimit c) ↔
    ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
tactic: intro F c α i e hα
state_before: 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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
⊢ Nonempty (IsColimit c) ↔
    ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
state_after: 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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
⊢ Nonempty (IsColimit c) →
    ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
tactic: refine ⟨?_, isUniversalColimit_extendCofan f t₁.isUniversal t₂.isUniversal c α i e hα⟩
state_before: 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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
⊢ Nonempty (IsColimit c) →
    ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
state_after: 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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)
⊢ IsPullback (c.ι.app { as := j }) (α.app { as := j }) i ((extendCofan c₁ c₂).ι.app { as := j })
tactic: intro ⟨Hc⟩ ⟨j⟩
state_before: 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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)
⊢ IsPullback (c.ι.app { as := j }) (α.app { as := j }) i ((extendCofan c₁ c₂).ι.app { as := j })
state_after: 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ })).ι.app j)
      ((mapPair (α.app { as := 0 }) (Sigma.desc fun b => α.app { as := b.succ } ≫ c₁.inj 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ } ≫ c₁.inj b) ≫ c₂.ι =
    (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b => c.ι.app { as := b.succ })).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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ }))
tactic: 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 ⟨?_⟩
state_before: 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ })).ι.app j)
      ((mapPair (α.app { as := 0 }) (Sigma.desc fun b => α.app { as := b.succ } ≫ c₁.inj 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ } ≫ c₁.inj b) ≫ c₂.ι =
    (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b => c.ι.app { as := b.succ })).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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ }))
state_after: 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ } ≫ c₁.inj b) ≫ c₂.ι =
    (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b => c.ι.app { as := b.succ })).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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ }))

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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ })).ι.app j)
      ((mapPair (α.app { as := 0 }) (Sigma.desc fun b => α.app { as := b.succ } ≫ c₁.inj 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 })
tactic: rotate_left
state_before: 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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ })).ι.app j)
      ((mapPair (α.app { as := 0 }) (Sigma.desc fun b => α.app { as := b.succ } ≫ c₁.inj 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 })
state_after: case refine_3.zero
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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ })).ι.app j)
      ((mapPair (α.app { as := 0 }) (Sigma.desc fun b => α.app { as := b.succ } ≫ c₁.inj b)).app j) i (c₂.ι.app j)
⊢ IsPullback (c.ι.app { as := 0 }) (α.app { as := 0 }) i ((extendCofan c₁ c₂).ι.app { as := 0 })

case refine_3.succ
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.306779, u_1} K
D : Type u_2
inst✝² : Category.{?u.306786, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr 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 := b.succ })).ι.app j)
      ((mapPair (α.app { as := 0 }) (Sigma.desc fun b => α.app { as := b.succ } ≫ c₁.inj b)).app j) i (c₂.ι.app j)
j : Fin n
a✝ :
  IsPullback (c.ι.app { as := j.castSucc }) (α.app { as := j.castSucc }) i
    ((extendCofan c₁ c₂).ι.app { as := j.castSucc })
⊢ IsPullback (c.ι.app { as := j.succ }) (α.app { as := j.succ }) i ((extendCofan c₁ c₂).ι.app { as := j.succ })
tactic: induction' j using Fin.inductionOn with j _