/-- If `X` is a cofiltered limit of profinite sets, then any clopen subset of `X` arises from
a clopen set in one of the terms in the limit.
-/
theorem exists_isClopen_of_cofiltered {U : Set C.pt} (hC : IsLimit C) (hU : IsClopen U) :
    ∃ (j : J) (V : Set (F.obj j)), IsClopen V ∧ U = C.π.app j ⁻¹' V := by
  have := preserves_smaller_limits_toTopCat.{u, v}
  -- First, we have the topological basis of the cofiltered limit obtained by pulling back
  -- clopen sets from the factors in the limit. By continuity, all such sets are again clopen.
  have hB := TopCat.isTopologicalBasis_cofiltered_limit.{u, v} (F ⋙ Profinite.toTopCat)
      (Profinite.toTopCat.mapCone C) (isLimitOfPreserves _ hC) (fun j => {W | IsClopen W}) ?_
      (fun i => isClopen_univ) (fun i U1 U2 hU1 hU2 => hU1.inter hU2) ?_
  rotate_left
  · intro i
    change TopologicalSpace.IsTopologicalBasis {W : Set (F.obj i) | IsClopen W}
    apply isTopologicalBasis_isClopen
  · rintro i j f V (hV : IsClopen _)
    exact ⟨hV.1.preimage ((F ⋙ toTopCat).map f).continuous,
      hV.2.preimage ((F ⋙ toTopCat).map f).continuous⟩
    -- Porting note: `<;> continuity` fails
  -- Using this, since `U` is open, we can write `U` as a union of clopen sets all of which
  -- are preimages of clopens from the factors in the limit.
  obtain ⟨S, hS, h⟩ := hB.open_eq_sUnion hU.1
  clear hB
  let j : S → J := fun s => (hS s.2).choose
  let V : ∀ s : S, Set (F.obj (j s)) := fun s => (hS s.2).choose_spec.choose
  have hV : ∀ s : S, IsClopen (V s) ∧ s.1 = C.π.app (j s) ⁻¹' V s := fun s =>
    (hS s.2).choose_spec.choose_spec

  -- Since `U` is also closed, hence compact, it is covered by finitely many of the
  -- clopens constructed in the previous step.
  have hUo : ∀ (i : ↑S), IsOpen ((fun s ↦ (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)
  · intro s
    exact (hV s).1.1.preimage (C.π.app (j s)).continuous
  have hsU : U ⊆ ⋃ (i : ↑S), (fun s ↦ (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i
  · dsimp only
    rw [h]
    rintro x ⟨T, hT, hx⟩
    refine' ⟨_, ⟨⟨T, hT⟩, rfl⟩, _⟩
    dsimp only [forget_map_eq_coe]
    rwa [← (hV ⟨T, hT⟩).2]
  have := hU.2.isCompact.elim_finite_subcover (fun s : S => C.π.app (j s) ⁻¹' V s) hUo hsU
  -- Porting note: same remark as after `hB`
  -- We thus obtain a finite set `G : Finset J` and a clopen set of `F.obj j` for each
  -- `j ∈ G` such that `U` is the union of the preimages of these clopen sets.
  obtain ⟨G, hG⟩ := this
  -- Since `J` is cofiltered, we can find a single `j0` dominating all the `j ∈ G`.
  -- Pulling back all of the sets from the previous step to `F.obj j0` and taking a union,
  -- we obtain a clopen set in `F.obj j0` which works.
  obtain ⟨j0, hj0⟩ := IsCofiltered.inf_objs_exists (G.image j)
  let f : ∀ s ∈ G, j0 ⟶ j s := fun s hs => (hj0 (Finset.mem_image.mpr ⟨s, hs, rfl⟩)).some
  let W : S → Set (F.obj j0) := fun s => if hs : s ∈ G then F.map (f s hs) ⁻¹' V s else Set.univ
  -- Conclude, using the `j0` and the clopen set of `F.obj j0` obtained above.
  refine' ⟨j0, ⋃ (s : S) (_ : s ∈ G), W s, _, _⟩
  · 

Proof state:
case refine_3.intro.intro.intro.intro.refine'_1
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ ProfiniteMax
C : Cone F
U : Set ↑C.pt.toCompHaus.toTop
hC : IsLimit C
hU : IsClopen U
this : PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} toTopCat
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS : S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}
h : U = ⋃₀ S
j : ↑S → J :=
  fun s =>
    Exists.choose (_ : ↑s ∈ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V})
V : (s : ↑S) → Set ↑(F.obj (j s)).toCompHaus.toTop :=
  fun s =>
    Exists.choose
      (_ :
        ∃
          V ∈
            (fun j => {W | IsClopen W})
              (Exists.choose
                (_ : ↑s ∈ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V})),
          ↑s =
            ⇑((toTopCat.mapCone C).π.app
                  (Exists.choose
                    (_ :
                      ↑s ∈
                        {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑((toTopCat.mapCone C).π.app j) ⁻¹' V}))) ⁻¹'
              V)
hV : ∀ (s : ↑S), IsClopen (V s) ∧ ↑s = ⇑(C.π.app (j s)) ⁻¹' V s
hUo : ∀ (i : ↑S), IsOpen ((fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i)
hsU : U ⊆ ⋃ i, (fun s => (forget Profinite).map (C.π.app (j s)) ⁻¹' V s) i
G : Finset ↑S
hG : U ⊆ ⋃ i ∈ G, (fun s => ⇑(C.π.app (j s)) ⁻¹' V s) i
j0 : J
hj0 : ∀ {X : J}, X ∈ Finset.image j G → Nonempty (j0 ⟶ X)
f : (s : ↑S) → s ∈ G → (j0 ⟶ j s) := fun s hs => Nonempty.some (_ : Nonempty (j0 ⟶ j s))
W : ↑S → Set ↑(F.obj j0).toCompHaus.toTop := fun s => if hs : s ∈ G then ⇑(F.map (f s hs)) ⁻¹' V s else Set.univ
⊢ IsClopen (⋃ s ∈ G, W s)

Next tactic:
apply isClopen_biUnion_finset