∀ {I : Type u_1} {C : Set (I → Bool)} [inst : (s : Finset I) → (i : I) → Decidable (i ∈ s)] (hC : IsCompact C)
  ⦃X Y : (Finset I)ᵒᵖ⦄ (f : X ⟶ Y),
  CategoryTheory.CategoryStruct.comp (((CategoryTheory.Functor.const (Finset I)ᵒᵖ).obj (Profinite.of ↑C)).map f)
      (CategoryTheory.ConcreteCategory.ofHom
        { toFun := Profinite.NobelingProof.ProjRestrict C fun x => x ∈ Opposite.unop Y, continuous_toFun := ⋯ }) =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.ConcreteCategory.ofHom
        { toFun := Profinite.NobelingProof.ProjRestrict C fun x => x ∈ Opposite.unop X, continuous_toFun := ⋯ })
      ((Profinite.NobelingProof.spanFunctor hC).map f)