-- Module: Mathlib.Topology.Category.TopCat.Limits.Konig
Goal:
J : Type u
inst : SmallCategory J
F : J ⥤ TopCat
inst_1 : IsCofilteredOrEmpty J
inst_2 : ∀ (j : J), Nonempty (↑(F.obj j) : Type (max u v))
inst_3 : ∀ (j : J), CompactSpace (↑(F.obj j) : Type (max u v))
inst_4 : ∀ (j : J), T2Space (↑(F.obj j) : Type (max u v))
obtain :
  ∀ (u : (j : J) → (↑(F.obj j) : Type (max u v)))
    (hu :
      Membership.mem (γ := Set ((j : J) → (↑(F.obj j) : Type (max u v))))
        (⋂ (i : FiniteDiagram J), partialSections F (i.snd (β := fun (G : Finset J) ↦ Finset (FiniteDiagramArrow G))))
        u),
    Nonempty (↑(limitCone.{u, v, u} F).pt : Type (max u v))
property :
  ∀ (u : (j : J) → (↑(F.obj j) : Type (max u v)))
    (hu :
      Membership.mem (γ := Set ((j : J) → (↑(F.obj j) : Type (max u v))))
        (⋂ (i : FiniteDiagram J), partialSections F (i.snd (β := fun (G : Finset J) ↦ Finset (FiniteDiagramArrow G))))
        u),
    Membership.mem (γ := Set ((j : J) → (↑(F.obj j) : Type (max v u))))
      {u : (j : J) → (↑(F.obj j) : Type (max v u)) |
        ∀ {i j : J} (f : i ⟶ j),
          (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC :=
                  carrier) (F.map f) :
                (a : (↑(F.obj i) : Type (max v u))) → (↑(F.obj j) : Type (max v u)))
              (u i) =
            u j}
      u
property₁ :
  ∀ (u : (j : J) → (↑(F.obj j) : Type (max u v)))
    (hu :
      Membership.mem (γ := Set ((j : J) → (↑(F.obj j) : Type (max u v))))
        (⋂ (i : FiniteDiagram J), partialSections F (i.snd (β := fun (G : Finset J) ↦ Finset (FiniteDiagramArrow G))))
        u)
    {X Y : J} (f : X ⟶ Y),
    (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
            (F.map f) :
          (a : (↑(F.obj X) : Type (max v u))) → (↑(F.obj Y) : Type (max v u)))
        (u X) =
      u Y
property₂ :
  ∀ (u : (j : J) → (↑(F.obj j) : Type (max u v)))
    (hu :
      Membership.mem (γ := Set ((j : J) → (↑(F.obj j) : Type (max u v))))
        (⋂ (i : FiniteDiagram J), partialSections F (i.snd (β := fun (G : Finset J) ↦ Finset (FiniteDiagramArrow G))))
        u)
    {X Y : J} (f : X ⟶ Y),
    (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
            (F.map f) :
          (a : (↑(F.obj X) : Type (max v u))) → (↑(F.obj Y) : Type (max v u)))
        (u X) =
      u Y
⊢ Nonempty (↑(limitCone.{u, v, u} F).pt : Type (max u v))

Tactic:
classical
  obtain ⟨u, hu⟩ :=
    IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed (fun G => partialSections F _)
      (partialSections.directed F) (fun G => partialSections.nonempty F _)
      (fun G => IsClosed.isCompact (partialSections.closed F _)) fun G =>
      partialSections.closed F _
  use u
  intro X Y f
  let G : FiniteDiagram J := ⟨{X, Y}, {⟨X, Y, by grind, by grind, f⟩}⟩
  exact hu _ ⟨G, rfl⟩ (Finset.mem_singleton_self _)

Premises used: IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed, TopCat.partialSections, IsClosed.isCompact, TopCat.partialSections.closed, Sigma.snd, TopCat.partialSections.directed, TopCat.partialSections.nonempty