Module: Mathlib.CategoryTheory.Limits.VanKampen

Goal state before:
pp: 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
⊢ IsVanKampenColimit (extendCofan c₁ c₂)

Tactic:
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 ?_ (.of_discrete _)).mp ⟨?_⟩
  rotate_left
  · ext ⟨⟨⟩⟩
    · simpa only [pair_obj_left, Functor.const_obj_obj, pair_obj_right, Discrete.functor_obj,
        NatTrans.comp_app, mapPair_left, BinaryCofan.ι_app_left, BinaryCofan.mk_pt,
        BinaryCofan.mk_inl, Functor.const_map_app, extendCofan_pt,
        extendCofan_ι_app, Fin.cases_zero] using! congr_app e ⟨0⟩
    · dsimp
      ext j
      simpa only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_ι_app,
        Category.assoc, extendCofan_pt, Functor.const_obj_obj, NatTrans.comp_app, extendCofan_ι_app,
        Fin.cases_succ, Functor.const_map_app] using! congr_app e ⟨j.succ⟩
  · let F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
    have : F = Discrete.functor F' := by
      apply Functor.hext
      · exact fun i ↦ rfl
      · rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
        simp [F']
    clear_value F'
    subst this
    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))))
    · intro T f₁ f₂
      simp only [Discrete.functor_obj, pair_obj_left, BinaryCofan.mk_pt, Functor.const_obj_obj,
        BinaryCofan.mk_inl, IsColimit.fac, Cofan.mk_pt, Cofan.mk_ι_app,
        Fin.cases_zero]
    · intro T f₁ f₂
      simp only [Discrete.functor_obj, pair_obj_right, BinaryCofan.mk_pt, Functor.const_obj_obj,
        BinaryCofan.mk_inr]
      ext j
      simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt,
        Cofan.mk_ι_app, IsColimit.fac, Fin.cases_succ]
    · intro T f₁ f₂ f₃ m₁ m₂
      simp only [Discrete.functor_obj_eq_as, pair_obj_left, BinaryCofan.mk_pt, const_obj_obj,
        BinaryCofan.mk_inl, pair_obj_right, BinaryCofan.mk_inr] at m₁ m₂ ⊢
      refine Hc.uniq (Cofan.mk T (Fin.cases f₁
        (fun i ↦ Sigma.ι (fun (j : Fin n) ↦ (Discrete.functor F').obj ⟨j.succ⟩) _ ≫ f₂))) _ ?_
      intro ⟨j⟩
      simp only [Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_ι_app]
      induction j using Fin.inductionOn
      · simp only [Fin.cases_zero, m₁]
      · simp only [← m₂, colimit.ι_desc_assoc, Discrete.functor_obj,
          Cofan.mk_pt, Cofan.mk_ι_app, Fin.cases_succ]
  induction j using Fin.inductionOn with
  | zero => exact t₂' ⟨WalkingPair.left⟩
  | succ j _ =>
    have t₁' := (@t₁ (Discrete.functor (fun j ↦ F.obj ⟨j.succ⟩)) (Cofan.mk _ _) (Discrete.natTrans
      fun i ↦ α.app _) (Sigma.desc (fun j ↦ α.app _ ≫ c₁.inj _)) ?_
      (.of_discrete _)).mp ⟨coproductIsCoproduct _⟩ ⟨j⟩
    rotate_left
    · ext ⟨j⟩
      dsimp
      rw [colimit.ι_desc]
      rfl
    simpa [Functor.const_obj_obj, Discrete.functor_obj, extendCofan_pt, extendCofan_ι_app,
      Fin.cases_succ, BinaryCofan.mk_pt, colimit.cocone_x, Cofan.mk_pt, Cofan.mk_ι_app,
      BinaryCofan.ι_app_right, BinaryCofan.mk_inr, colimit.ι_desc,
      Discrete.natTrans_app] using! t₁'.paste_horiz (t₂' ⟨WalkingPair.right⟩)

Goal state after: