/-- Given a *compatible* collection of topological bases for the factors in a cofiltered limit
which contain `Set.univ` and are closed under intersections, the induced *naive* collection
of sets in the limit is, in fact, a topological basis.
-/
theorem isTopologicalBasis_cofiltered_limit (T : ∀ j, Set (Set (F.obj j)))
    (hT : ∀ j, IsTopologicalBasis (T j)) (univ : ∀ i : J, Set.univ ∈ T i)
    (inter : ∀ (i) (U1 U2 : Set (F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i)
    (compat : ∀ (i j : J) (f : i ⟶ j) (V : Set (F.obj j)) (_hV : V ∈ T j), F.map f ⁻¹' V ∈ T i) :
    IsTopologicalBasis
      {U : Set C.pt | ∃ (j : _) (V : Set (F.obj j)), V ∈ T j ∧ U = C.π.app j ⁻¹' V} := by
  classical
  -- The limit cone for `F` whose topology is defined as an infimum.
  let D := limitConeInfi F
  -- The isomorphism between the cone point of `C` and the cone point of `D`.
  let E : C.pt ≅ D.pt := hC.conePointUniqueUpToIso (limitConeInfiIsLimit _)
  have hE : Inducing E.hom := (TopCat.homeoOfIso E).inducing
  -- Reduce to the assertion of the theorem with `D` instead of `C`.
  suffices
    IsTopologicalBasis
      {U : Set D.pt | ∃ (j : _) (V : Set (F.obj j)), V ∈ T j ∧ U = D.π.app j ⁻¹' V} by
    convert this.inducing hE
    ext U0
    constructor
    · rintro ⟨j, V, hV, rfl⟩
      refine' ⟨D.π.app j ⁻¹' V, ⟨j, V, hV, rfl⟩, rfl⟩
    · rintro ⟨W, ⟨j, V, hV, rfl⟩, rfl⟩
      refine' ⟨j, V, hV, rfl⟩
  -- Using `D`, we can apply the characterization of the topological basis of a
  -- topology defined as an infimum...
  convert isTopologicalBasis_iInf hT fun j (x : D.pt) => D.π.app j x using 1
  ext U0
  constructor
  · rintro ⟨j, V, hV, rfl⟩
    let U : ∀ i, Set (F.obj i) := fun i => if h : i = j then by rw [h]; exact V else Set.univ
    refine' ⟨U, {j}, _, _⟩
    · rintro i h
      rw [Finset.mem_singleton] at h
      dsimp
      rw [dif_pos h]
      subst h
      exact hV
    · dsimp
      simp
  · rintro ⟨U, G, h1, h2⟩
    obtain ⟨j, hj⟩ := IsCofiltered.inf_objs_exists G
    let g : ∀ (e) (_he : e ∈ G), j ⟶ e := fun _ he => (hj he).some
    let Vs : J → Set (F.obj j) := fun e => if h : e ∈ G then F.map (g e h) ⁻¹' U e else Set.univ
    let V : Set (F.obj j) := ⋂ (e : J) (_he : e ∈ G), Vs e
    refine' ⟨j, V, _, _⟩
    · -- An intermediate claim used to apply induction along `G : Finset J` later on.
      have :
        ∀ (S : Set (Set (F.obj j))) (E : Finset J) (P : J → Set (F.obj j)) (_univ : Set.univ ∈ S)
          (_inter : ∀ A B : Set (F.obj j), A ∈ S → B ∈ S → A ∩ B ∈ S)
          (_cond : ∀ (e : J) (_he : e ∈ E), P e ∈ S), (⋂ (e) (_he : e ∈ E), P e) ∈ S := by
        intro S E
        induction E using Finset.induction_on with
        | empty =>
          intro P he _hh
          simpa
        | @insert a E _ha hh1 =>
          intro hh2 hh3 hh4 hh5
          rw [Finset.set_biInter_insert]
          refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _)
          intro e he
          exact hh5 e (Finset.mem_insert_of_mem he)
      -- use the intermediate claim to finish off the goal using `univ` and `inter`.
      refine' this _ _ _ (univ _) (inter _) _
      intro e he
      dsimp
      

Proof state:
case h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ TopCatMax
C : Cone F
hC : IsLimit C
T : (j : J) → Set (Set ↑(F.obj j))
hT : ∀ (j : J), IsTopologicalBasis (T j)
univ : ∀ (i : J), Set.univ ∈ T i
inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i
compat : ∀ (i j : J) (f : i ⟶ j), ∀ V ∈ T j, ⇑(F.map f) ⁻¹' V ∈ T i
D : Cone F := limitConeInfi F
E : C.pt ≅ D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)
hE : Inducing ⇑E.hom
U0 : Set ↑D.pt
U : (i : J) → Set ↑(F.obj i)
G : Finset J
h1 : ∀ i ∈ G, U i ∈ T i
h2 : U0 = ⋂ i ∈ G, (fun x => (D.π.app i) x) ⁻¹' U i
j : J
hj : ∀ {X : J}, X ∈ G → Nonempty (j ⟶ X)
g : (e : J) → e ∈ G → (j ⟶ e) := fun x he => Nonempty.some (_ : Nonempty (j ⟶ x))
Vs : J → Set ↑(F.obj j) := fun e => if h : e ∈ G then ⇑(F.map (g e h)) ⁻¹' U e else Set.univ
V : Set ↑(F.obj j) := ⋂ e ∈ G, Vs e
this :
  ∀ (S : Set (Set ↑(F.obj j))) (E : Finset J) (P : J → Set ↑(F.obj j)),
    Set.univ ∈ S → (∀ (A B : Set ↑(F.obj j)), A ∈ S → B ∈ S → A ∩ B ∈ S) → (∀ e ∈ E, P e ∈ S) → ⋂ e ∈ E, P e ∈ S
e : J
he : e ∈ G
⊢ (if h : e ∈ G then ⇑(F.map (Nonempty.some (_ : Nonempty (j ⟶ e)))) ⁻¹' U e else Set.univ) ∈ T j

Next tactic:
rw [dif_pos he]