-- Module: Mathlib.Topology.Category.CompHausLike.Cartesian
Goal:
P : TopCat.{u} → Prop
X Y : CompHausLike P
inst : HasProp P ((↑(X.toTop (P := P)) : Type u) × (↑(Y.toTop (P := P)) : Type u))
x : BinaryFan X Y
x_1 : x.pt ⟶ of P ((↑(X.toTop (P := P)) : Type u) × (↑(Y.toTop (P := P)) : Type u))
h₁ : x_1 ≫ ofHom P { toFun := Prod.fst, continuous_toFun := ⋯ } = x.fst
h₂ : x_1 ≫ ofHom P { toFun := Prod.snd, continuous_toFun := ⋯ } = x.snd
h :
  ∀ (x_2 : (↑(x.pt.toTop (P := P)) : Type u)),
    Eq.{u + 1} (α := (↑(X.toTop (P := P)) : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
              C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u))) (CC := fun (X : CompHausLike P) ↦
              (↑(X.toTop (P := P)) : Type u)) x_1 :
            (a : (↑x.pt.toTop : Type u)) → (↑(of P ((↑X.toTop : Type u) × (↑Y.toTop : Type u))).toTop : Type u))
          x_2).1
      ((@ConcreteCategory.hom _ (category P)
              (fun (x1 x2 : CompHausLike P) ↦ C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u)))
              (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) _ _ x.pt
              (of P ((↑(X.toTop (P := P)) : Type u) × (↑(Y.toTop (P := P)) : Type u)))
              ((fun (s : BinaryFan X Y) ↦
                  (ofHom P
                      {
                        toFun := fun (x : (↑s.1.1 : Type u)) ↦
                          ((ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
                                  C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u))) (CC :=
                                  fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) s.fst :
                                (a :
                                    (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u))
                                      (((Functor.const (Discrete WalkingPair)).obj s.pt).obj
                                        { as := WalkingPair.left })) →
                                  (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u))
                                    ((pair X Y).obj { as := WalkingPair.left }))
                              x,
                            (ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
                                  C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u))) (CC :=
                                  fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) s.snd :
                                (a :
                                    (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u))
                                      (((Functor.const (Discrete WalkingPair)).obj s.pt).obj
                                        { as := WalkingPair.right })) →
                                  (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u))
                                    ((pair X Y).obj { as := WalkingPair.right }))
                              x),
                        continuous_toFun := ⋯ } :
                    of P (↑s.1.1 : Type u) ⟶ of P ((↑(X.toTop (P := P)) : Type u) × (↑(Y.toTop (P := P)) : Type u))))
                x) :
            (a : (↑x.pt.toTop : Type u)) → (↑(of P ((↑X.toTop : Type u) × (↑Y.toTop : Type u))).toTop : Type u))
          x_2).1
h₃ :
  ∀ (x_2 : (↑(x.pt.toTop (P := P)) : Type u)),
    Eq.{u + 1} (α := (↑(Y.toTop (P := P)) : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
              C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u))) (CC := fun (X : CompHausLike P) ↦
              (↑(X.toTop (P := P)) : Type u)) x_1 :
            (a : (↑x.pt.toTop : Type u)) → (↑(of P ((↑X.toTop : Type u) × (↑Y.toTop : Type u))).toTop : Type u))
          x_2).2
      ((@ConcreteCategory.hom _ (category P)
              (fun (x1 x2 : CompHausLike P) ↦ C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u)))
              (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) _ _ x.pt
              (of P ((↑(X.toTop (P := P)) : Type u) × (↑(Y.toTop (P := P)) : Type u)))
              ((fun (s : BinaryFan X Y) ↦
                  (ofHom P
                      {
                        toFun := fun (x : (↑s.1.1 : Type u)) ↦
                          ((ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
                                  C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u))) (CC :=
                                  fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) s.fst :
                                (a :
                                    (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u))
                                      (((Functor.const (Discrete WalkingPair)).obj s.pt).obj
                                        { as := WalkingPair.left })) →
                                  (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u))
                                    ((pair X Y).obj { as := WalkingPair.left }))
                              x,
                            (ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
                                  C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u))) (CC :=
                                  fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) s.snd :
                                (a :
                                    (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u))
                                      (((Functor.const (Discrete WalkingPair)).obj s.pt).obj
                                        { as := WalkingPair.right })) →
                                  (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u))
                                    ((pair X Y).obj { as := WalkingPair.right }))
                              x),
                        continuous_toFun := ⋯ } :
                    of P (↑s.1.1 : Type u) ⟶ of P ((↑(X.toTop (P := P)) : Type u) × (↑(Y.toTop (P := P)) : Type u))))
                x) :
            (a : (↑x.pt.toTop : Type u)) → (↑(of P ((↑X.toTop : Type u) × (↑Y.toTop : Type u))).toTop : Type u))
          x_2).2
⊢ x_1 =
    (fun (s : BinaryFan X Y) ↦
        (ofHom P
            {
              toFun := fun (x : (↑s.1.1 : Type u)) ↦
                ((ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
                        C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u))) (CC :=
                        fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) s.fst :
                      (a :
                          (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u))
                            (((Functor.const (Discrete WalkingPair)).obj s.pt).obj { as := WalkingPair.left })) →
                        (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) ((pair X Y).obj { as := WalkingPair.left }))
                    x,
                  (ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
                        C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u))) (CC :=
                        fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) s.snd :
                      (a :
                          (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u))
                            (((Functor.const (Discrete WalkingPair)).obj s.pt).obj { as := WalkingPair.right })) →
                        (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) ((pair X Y).obj { as := WalkingPair.right }))
                    x),
              continuous_toFun := ⋯ } :
          of P (↑s.1.1 : Type u) ⟶ of P ((↑(X.toTop (P := P)) : Type u) × (↑(Y.toTop (P := P)) : Type u))))
      x

Tactic:
ext x