∀ (X : LightProfinite) [inst : Finite ↑X.toTop]
  (x : CategoryTheory.Limits.Cofan fun x => LightProfinite.of PUnit.{u_1 + 1})
  (x_1 : (LightCondensed.fintypeCatAsCofan X).pt ⟶ x.pt),
  (∀ (j : ↑X.toTop), CategoryTheory.CategoryStruct.comp ((LightCondensed.fintypeCatAsCofan X).inj j) x_1 = x.inj j) →
    x_1 =
      (fun t =>
          CategoryTheory.ConcreteCategory.ofHom
            { toFun := fun x => (CategoryTheory.ConcreteCategory.hom (t.inj x)) PUnit.unit, continuous_toFun := ⋯ })
        x