-- Module: Mathlib.CategoryTheory.Limits.VanKampen
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (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 ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
H₁ :
  IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ })
    (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
      pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯)
refine_4 :
  Eq (α :=
    obj.{0, v, 0, u} (pair (F.obj { as := 0 : Discrete (Fin (n + 1)) }) (pullback c₂.inr i))
        { as := WalkingPair.left : Discrete WalkingPair } ⟶
      obj.{0, v, 0, u} (obj (const (Discrete WalkingPair)) c₂.pt) { as := WalkingPair.left : Discrete WalkingPair })
    (NatTrans.app (mapPair (α.app { as := 0 }) (pullback.fst c₂.inr i) ≫ c₂.ι) { as := WalkingPair.left })
    (NatTrans.app
      (CategoryStruct.comp (Y :=
        obj (const (Discrete WalkingPair))
          (Cocone.pt (F := pair (F.obj { as := 0 : Discrete (Fin (n + 1)) }) (pullback c₂.inr i))
            (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))))
        (Cocone.ι (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i)))
        (map (const (Discrete WalkingPair)) i))
      { as := WalkingPair.left })
refine_4₁ :
  Eq (α :=
    obj.{0, v, 0, u} (pair (F.obj { as := 0 : Discrete (Fin (n + 1)) }) (pullback c₂.inr i))
        { as := WalkingPair.right : Discrete WalkingPair } ⟶
      obj.{0, v, 0, u} (obj (const (Discrete WalkingPair)) c₂.pt) { as := WalkingPair.right : Discrete WalkingPair })
    (NatTrans.app (mapPair (α.app { as := 0 }) (pullback.fst c₂.inr i) ≫ c₂.ι) { as := WalkingPair.right })
    (NatTrans.app
      (CategoryStruct.comp (Y :=
        obj (const (Discrete WalkingPair))
          (Cocone.pt (F := pair (F.obj { as := 0 : Discrete (Fin (n + 1)) }) (pullback c₂.inr i))
            (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))))
        (Cocone.ι (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i)))
        (map (const (Discrete WalkingPair)) i))
      { as := WalkingPair.right })
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{0, v, 0, u} (Discrete WalkingPair) C) (pair (F.obj { as := 0 }) (pullback c₂.inr i))
      (obj (const (Discrete WalkingPair)) c₂.pt))
    (mapPair (α.app { as := 0 }) (pullback.fst c₂.inr i) ≫ c₂.ι)
    (CategoryStruct.comp (Y :=
      obj (const (Discrete WalkingPair))
        (Cocone.pt (F := pair (F.obj { as := 0 : Discrete (Fin (n + 1)) }) (pullback c₂.inr i))
          (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))))
      (Cocone.ι (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i)))
      (map (const (Discrete WalkingPair)) i))

Tactic:
ext ⟨⟨⟩⟩