instance types.finitaryExtensive : FinitaryExtensive (Type u) := by
  classical
  rw [finitaryExtensive_iff_of_isTerminal (Type u) PUnit Types.isTerminalPunit _
      (Types.binaryCoproductColimit _ _)]
  apply BinaryCofan.isVanKampen_mk _ _ (fun X Y => Types.binaryCoproductColimit X Y) _
      fun f g => (Limits.Types.pullbackLimitCone f g).2
  · intros _ _ _ _ f hαX hαY
    constructor
    · refine' ⟨⟨hαX.symm⟩, ⟨PullbackCone.isLimitAux' _ _⟩⟩
      intro s
      have : ∀ x, ∃! y, s.fst x = Sum.inl y := by
        intro x
        cases' h : s.fst x with val val
        · simp only [Types.binaryCoproductCocone_pt, Functor.const_obj_obj, Sum.inl.injEq,
            exists_unique_eq']
        · apply_fun f at h
          cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαY val : _).symm
      delta ExistsUnique at this
      choose l hl hl' using this
      exact ⟨l, (funext hl).symm, Types.isTerminalPunit.hom_ext _ _,
        fun {l'} h₁ _ => funext fun x => hl' x (l' x) (congr_fun h₁ x).symm⟩
    · refine' ⟨⟨hαY.symm⟩, ⟨PullbackCone.isLimitAux' _ _⟩⟩
      intro s
      have : ∀ x, ∃! y, s.fst x = Sum.inr y := by
        intro x
        cases' h : s.fst x with val val
        · apply_fun f at h
          cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun hαX val : _).symm
        · simp only [Types.binaryCoproductCocone_pt, Functor.const_obj_obj, Sum.inr.injEq,
            exists_unique_eq']
      delta ExistsUnique at this
      choose l hl hl' using this
      

Proof state:
case h₁.right
J : Type v'
inst✝² : Category.{u', v'} J
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u''
inst✝ : Category.{v'', u''} D
X Y : C
X'✝ Y'✝ : Type u
αX✝ : X'✝ ⟶ PUnit.{u + 1}
αY✝ : Y'✝ ⟶ PUnit.{u + 1}
f : (Types.binaryCoproductCocone X'✝ Y'✝).pt ⟶ (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt
hαX :
  αX✝ ≫ BinaryCofan.inl (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}) =
    BinaryCofan.inl (Types.binaryCoproductCocone X'✝ Y'✝) ≫ f
hαY :
  αY✝ ≫ BinaryCofan.inr (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}) =
    BinaryCofan.inr (Types.binaryCoproductCocone X'✝ Y'✝) ≫ f
s : PullbackCone f (BinaryCofan.inr (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}))
l : s.pt → Y'✝
hl : ∀ (x : s.pt), (fun y => PullbackCone.fst s x = Sum.inr y) (l x)
hl' : ∀ (x : s.pt) (y : Y'✝), (fun y => PullbackCone.fst s x = Sum.inr y) y → y = l x
⊢ { l //
    l ≫
          PullbackCone.fst
            (PullbackCone.mk (BinaryCofan.inr (Types.binaryCoproductCocone X'✝ Y'✝)) αY✝
              (_ :
                BinaryCofan.inr (Types.binaryCoproductCocone X'✝ Y'✝) ≫ f =
                  αY✝ ≫ BinaryCofan.inr (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}))) =
        PullbackCone.fst s ∧
      l ≫
            PullbackCone.snd
              (PullbackCone.mk (BinaryCofan.inr (Types.binaryCoproductCocone X'✝ Y'✝)) αY✝
                (_ :
                  BinaryCofan.inr (Types.binaryCoproductCocone X'✝ Y'✝) ≫ f =
                    αY✝ ≫ BinaryCofan.inr (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}))) =
          PullbackCone.snd s ∧
        ∀
          {m :
            s.pt ⟶
              (PullbackCone.mk (BinaryCofan.inr (Types.binaryCoproductCocone X'✝ Y'✝)) αY✝
                  (_ :
                    BinaryCofan.inr (Types.binaryCoproductCocone X'✝ Y'✝) ≫ f =
                      αY✝ ≫ BinaryCofan.inr (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}))).pt},
          m ≫
                PullbackCone.fst
                  (PullbackCone.mk (BinaryCofan.inr (Types.binaryCoproductCocone X'✝ Y'✝)) αY✝
                    (_ :
                      BinaryCofan.inr (Types.binaryCoproductCocone X'✝ Y'✝) ≫ f =
                        αY✝ ≫ BinaryCofan.inr (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}))) =
              PullbackCone.fst s →
            m ≫
                  PullbackCone.snd
                    (PullbackCone.mk (BinaryCofan.inr (Types.binaryCoproductCocone X'✝ Y'✝)) αY✝
                      (_ :
                        BinaryCofan.inr (Types.binaryCoproductCocone X'✝ Y'✝) ≫ f =
                          αY✝ ≫ BinaryCofan.inr (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}))) =
                PullbackCone.snd s →
              m = l }

Next tactic:
exact ⟨l, (funext hl).symm, Types.isTerminalPunit.hom_ext _ _,
        fun {l'} h₁ _ => funext fun x => hl' x (l' x) (congr_fun h₁ x).symm⟩