-- Lean 4 proof step from Mathlib.Topology.Category.CompHausLike.EffectiveEpi (goal)
Goal:
P : TopCat.{u} → Prop
B X : CompHausLike P
π : X ⟶ B
hπ :
  Function.Surjective (α := (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) X) (β :=
    (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) B)
    (⇑(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)) π) :
      (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X) → (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) B)
W : CompHausLike P
e : X ⟶ W
h : ∀ {Z : CompHausLike P} (g₁ g₂ : Z ⟶ X), g₁ ≫ π = g₂ ≫ π → g₁ ≫ e = g₂ ≫ e
g : B ⟶ W
hm : π ≫ g = e
⊢ g =
    ofHom P
      ((⋯.liftEquiv :
          (a :
              { g : C((fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X, (↑W.toTop : Type u)) //
                Function.FactorsThrough
                  (⇑g : (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X) → (↑W.toTop : Type u))
                  (⇑(ConcreteCategory.hom π) :
                    (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X) →
                      (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) B) }) →
            C((fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) B, (↑W.toTop : Type u)))
        (Subtype.mk (p :=
          fun (g : C((fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) X, (↑(W.toTop (P := P)) : Type u))) ↦
          Function.FactorsThrough (α := (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) X) (β :=
            (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) B) (γ := (↑(W.toTop (P := P)) : Type u))
            (⇑g : (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X) → (↑W.toTop : 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)) π) :
              (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X) →
                (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) B))
          (TopCat.Hom.hom e.hom) ⋯))
have h₁ : 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)) g =
  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 := B) (Y := W)
    (ofHom P
        ((sorry.liftEquiv :
            (a :
                { g : C((fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X, (↑W.toTop : Type u)) //
                  Function.FactorsThrough
                    (⇑g : (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X) → (↑W.toTop : Type u))
                    (⇑(ConcreteCategory.hom π) :
                      (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X) →
                        (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) B) }) →
              C((fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) B, (↑W.toTop : Type u)))
          (Subtype.mk (p :=
            fun (g : C((fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) X, (↑(W.toTop (P := P)) : Type u))) ↦
            Function.FactorsThrough (α := (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) X) (β :=
              (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) B) (γ := (↑(W.toTop (P := P)) : Type u))
              (⇑g : (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X) → (↑W.toTop : 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)) π) :
                (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X) →
                  (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) B))
            (TopCat.Hom.hom e.hom) sorry)) :
      of P ((fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) B) ⟶ of P (↑(W.toTop (P := P)) : Type u))
Tactic: apply ConcreteCategory.ext