-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.AsLimit (goal)
Goal:
X : Profinite
⊢ Function.Bijective (α :=
    (fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
        (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u))
      X.asLimitCone.pt)
    (β :=
    (fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
        (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u))
      (limitCone X.diagram).pt)
    (⇑(ConcreteCategory.hom (FC :=
          fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
          C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u),
            (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
          (CC := fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
          (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)) (X :=
          X.asLimitCone.pt) ((limitConeIsLimit X.diagram).lift X.asLimitCone)) :
      (a :
          (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
            X.asLimitCone.pt) →
        (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
          (limitCone X.diagram).pt)
have refine_1 : ∀
  (a b :
    (fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
        (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u))
      X.asLimitCone.pt)
  (h :
    Eq (α :=
      (fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
          (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u))
        (limitCone X.diagram).pt)
      ((ConcreteCategory.hom (FC :=
              fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
              C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u),
                (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
              (CC := fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
              (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)) (X :=
              X.asLimitCone.pt) ((limitConeIsLimit X.diagram).lift X.asLimitCone) :
            (a :
                (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
                  X.asLimitCone.pt) →
              (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
                (limitCone X.diagram).pt)
          a :
        (fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
            (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u))
          (limitCone X.diagram).pt)
      ((ConcreteCategory.hom (FC :=
              fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
              C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u),
                (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
              (CC := fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
              (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)) (X :=
              X.asLimitCone.pt) ((limitConeIsLimit X.diagram).lift X.asLimitCone) :
            (a :
                (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
                  X.asLimitCone.pt) →
              (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
                (limitCone X.diagram).pt)
          b :
        (fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
            (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u))
          (limitCone X.diagram).pt)),
  a = b
Tactic: refine ⟨fun a b h => ?_, fun a => ?_⟩