-- Module: Mathlib.Topology.Category.Profinite.Product
Goal:
ι : Type u
X : ι → Type
inst : (i : ι) → TopologicalSpace (X i)
C : Set ((i : ι) → X i)
J K : ι → Prop
inst_1 : ∀ (i : ι), T2Space (X i)
inst_2 : ∀ (i : ι), TotallyDisconnectedSpace (X i)
hC : IsCompact C
this : CompactSpace.{u} (↑C : Type u)
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 (indexFunctor (X := X) (C := C) hC)).pt
b : (↑C : Type u)
hb :
  ∀ (J : Finset ι),
    (π_app (X := X) C fun (x : ι) ↦ x ∈ J : (a : (↑C : Type u)) → (↑(obj C fun (x : ι) ↦ x ∈ J) : Type u)) b =
      (↑a : (j : (Finset ι)ᵒᵖ) → (↑(((indexFunctor hC ⋙ profiniteToCompHaus) ⋙ compHausToTop).obj j) : Type u)) (op J)
h :
  (↑((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 :=
              Cone.pt (F := indexFunctor (X := X) (C := C) hC) (indexCone (X := X) (C := C) hC))
              ((limitConeIsLimit (indexFunctor (X := X) (C := C) hC)).lift (indexCone (X := X) (C := C) hC)) :
            (a :
                (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
                  (indexCone hC).pt) →
              (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
                (limitCone (indexFunctor hC)).pt)
          b) :
      (j : (Finset ι)ᵒᵖ) → (↑(((indexFunctor hC ⋙ profiniteToCompHaus) ⋙ compHausToTop).obj j) : Type u)) =
    (↑a : (j : (Finset ι)ᵒᵖ) → (↑(((indexFunctor hC ⋙ profiniteToCompHaus) ⋙ compHausToTop).obj j) : Type u))
⊢ (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 :=
          Cone.pt (F := indexFunctor (X := X) (C := C) hC) (indexCone (X := X) (C := C) hC))
          ((limitConeIsLimit (indexFunctor (X := X) (C := C) hC)).lift (indexCone (X := X) (C := C) hC)) :
        (a :
            (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
              (indexCone hC).pt) →
          (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
            (limitCone (indexFunctor hC)).pt)
      b =
    a

Tactic:
apply Subtype.ext

Premises used: Subtype.ext