-- Lean 4 proof step from Mathlib.Topology.Category.LightProfinite.Limits (goal)
Goal:
X Y Z : LightProfinite
f : X ⟶ Z
g : Y ⟶ Z
h : Epi g
⊢ Epi (X :=
    CompHausLike.pullback (P := fun (X : TopCat.{u_1}) ↦
      TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) f g)
    (CompHausLike.pullback.fst (P := fun (X : TopCat.{u_1}) ↦
      TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) f g)
have rw : ∀
  (h :
    Function.Surjective (α :=
      (fun
            (X :
              CompHausLike fun (X : TopCat.{u_1}) ↦
                TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) ↦
          (↑(X.toTop (P := fun (X : TopCat.{u_1}) ↦
                TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
            Type u_1))
        Y)
      (β :=
      (fun
            (X :
              CompHausLike fun (X : TopCat.{u_1}) ↦
                TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) ↦
          (↑(X.toTop (P := fun (X : TopCat.{u_1}) ↦
                TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
            Type u_1))
        Z)
      (⇑(ConcreteCategory.hom (FC :=
            fun
              (x1 x2 :
                CompHausLike fun (X : TopCat.{u_1}) ↦
                  TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) ↦
            C((↑(x1.toTop (P := fun (X : TopCat.{u_1}) ↦
                    TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
                Type u_1),
              (↑(x2.toTop (P := fun (X : TopCat.{u_1}) ↦
                    TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
                Type u_1)))
            (CC :=
            fun
              (X :
                CompHausLike fun (X : TopCat.{u_1}) ↦
                  TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) ↦
            (↑(X.toTop (P := fun (X : TopCat.{u_1}) ↦
                  TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
              Type u_1))
            g) :
        (a :
            (fun
                  (X :
                    CompHausLike fun (X : TopCat) ↦
                      TotallyDisconnectedSpace (↑X : Type u_1) ∧ SecondCountableTopology (↑X : Type u_1)) ↦
                (↑X.toTop : Type u_1))
              Y) →
          (fun
                (X :
                  CompHausLike fun (X : TopCat) ↦
                    TotallyDisconnectedSpace (↑X : Type u_1) ∧ SecondCountableTopology (↑X : Type u_1)) ↦
              (↑X.toTop : Type u_1))
            Z)),
  Function.Surjective (α :=
    (fun
          (X :
            CompHausLike fun (X : TopCat.{u_1}) ↦
              TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) ↦
        (↑(X.toTop (P := fun (X : TopCat.{u_1}) ↦
              TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
          Type u_1))
      (CompHausLike.pullback (P := fun (X : TopCat.{u_1}) ↦
        TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) f g))
    (β :=
    (fun
          (X :
            CompHausLike fun (X : TopCat.{u_1}) ↦
              TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) ↦
        (↑(X.toTop (P := fun (X : TopCat.{u_1}) ↦
              TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
          Type u_1))
      X)
    (⇑(ConcreteCategory.hom (FC :=
          fun
            (x1 x2 :
              CompHausLike fun (X : TopCat.{u_1}) ↦
                TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) ↦
          C((↑(x1.toTop (P := fun (X : TopCat.{u_1}) ↦
                  TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
              Type u_1),
            (↑(x2.toTop (P := fun (X : TopCat.{u_1}) ↦
                  TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
              Type u_1)))
          (CC :=
          fun
            (X :
              CompHausLike fun (X : TopCat.{u_1}) ↦
                TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) ↦
          (↑(X.toTop (P := fun (X : TopCat.{u_1}) ↦
                TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1))) :
            Type u_1))
          (X :=
          CompHausLike.pullback (P := fun (X : TopCat.{u_1}) ↦
            TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) f g)
          (CompHausLike.pullback.fst (P := fun (X : TopCat.{u_1}) ↦
            TotallyDisconnectedSpace.{u_1} (↑X : Type u_1) ∧ SecondCountableTopology.{u_1} (↑X : Type u_1)) f g)) :
      (a :
          (fun
                (X :
                  CompHausLike fun (X : TopCat) ↦
                    TotallyDisconnectedSpace (↑X : Type u_1) ∧ SecondCountableTopology (↑X : Type u_1)) ↦
              (↑X.toTop : Type u_1))
            (CompHausLike.pullback f g)) →
        (fun
              (X :
                CompHausLike fun (X : TopCat) ↦
                  TotallyDisconnectedSpace (↑X : Type u_1) ∧ SecondCountableTopology (↑X : Type u_1)) ↦
            (↑X.toTop : Type u_1))
          X)
Tactic: rw [LightProfinite.epi_iff_surjective] at h ⊢
Cleared hypotheses: h