Module: Mathlib.CategoryTheory.Limits.VanKampen

Goal state before:
pp: 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
j : Fin (n + 1)
F' : Fin (n + 1) → C
c : Cocone (Discrete.functor F')
α : Discrete.functor F' ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
Hc : IsColimit c
⊢ IsColimit (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b ↦ c.ι.app { as := b.succ }))

Tactic:
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))))

Goal state after:
case refine_2.hd₁
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
j : Fin (n + 1)
F' : Fin (n + 1) → C
c : Cocone (Discrete.functor F')
α : Discrete.functor F' ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
Hc : IsColimit c
⊢ ∀ {T : C} (f : F' 0 ⟶ T) (g : (∐ fun j ↦ (Discrete.functor F').obj { as := j.succ }) ⟶ T),
    (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b ↦ c.ι.app { as := b.succ })).inl ≫
        Hc.desc
          (Cofan.mk T fun i ↦
            Fin.cases f (fun i ↦ Sigma.ι (fun j ↦ (Discrete.functor F').obj { as := j.succ }) i ≫ g) i) =
      f

case refine_2.hd₂
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
j : Fin (n + 1)
F' : Fin (n + 1) → C
c : Cocone (Discrete.functor F')
α : Discrete.functor F' ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
Hc : IsColimit c
⊢ ∀ {T : C} (f : F' 0 ⟶ T) (g : (∐ fun j ↦ (Discrete.functor F').obj { as := j.succ }) ⟶ T),
    (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b ↦ c.ι.app { as := b.succ })).inr ≫
        Hc.desc
          (Cofan.mk T fun i ↦
            Fin.cases f (fun i ↦ Sigma.ι (fun j ↦ (Discrete.functor F').obj { as := j.succ }) i ≫ g) i) =
      g

case refine_2.uniq
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
j : Fin (n + 1)
F' : Fin (n + 1) → C
c : Cocone (Discrete.functor F')
α : Discrete.functor F' ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
Hc : IsColimit c
⊢ ∀ {T : C} (f : F' 0 ⟶ T) (g : (∐ fun j ↦ (Discrete.functor F').obj { as := j.succ }) ⟶ T)
    (m : (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b ↦ c.ι.app { as := b.succ })).pt ⟶ T),
    (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b ↦ c.ι.app { as := b.succ })).inl ≫ m = f →
      (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b ↦ c.ι.app { as := b.succ })).inr ≫ m = g →
        m =
          Hc.desc
            (Cofan.mk T fun i ↦
              Fin.cases f (fun i ↦ Sigma.ι (fun j ↦ (Discrete.functor F').obj { as := j.succ }) i ≫ g) i)