-- Mathlib/CategoryTheory/Limits/VanKampen.lean
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)

Proof state:
case refine_3
C : Type u
inst✝² : Category.{v, u} C
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
C : Type u
inst✝² : Category.{v, u} C
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
C : Type u
inst✝² : Category.{v, u} C
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:
rotate_left

Result:
case refine_1
C : Type u
inst✝² : Category.{v, u} C
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
C : Type u
inst✝² : Category.{v, u} C
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
C : Type u
inst✝² : Category.{v, u} C
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 })

Proof state:
case refine_3.succ.refine_1
C : Type u
inst✝² : Category.{v, u} C
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 })
⊢ (Discrete.natTrans fun i => α.app { as := i.as.succ }) ≫ c₁.ι =
    (Cofan.mk (colimit.cocone (Discrete.functor fun j => F.obj { as := j.succ })).1
          (Sigma.ι fun j => F.obj { as := j.succ })).ι ≫
      (Functor.const (Discrete (Fin n))).map (Sigma.desc fun j => α.app { as := j.succ } ≫ c₁.inj j)

Tactic:
ext ⟨j⟩

Result:
case refine_3.succ.refine_1.w.h.mk
C : Type u
inst✝² : Category.{v, u} C
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 })
j : Fin n
⊢ ((Discrete.natTrans fun i => α.app { as := i.as.succ }) ≫ c₁.ι).app { as := j } =
    ((Cofan.mk (colimit.cocone (Discrete.functor fun j => F.obj { as := j.succ })).1
              (Sigma.ι fun j => F.obj { as := j.succ })).ι ≫
          (Functor.const (Discrete (Fin n))).map (Sigma.desc fun j => α.app { as := j.succ } ≫ c₁.inj j)).app
      { as := j }