∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (P : CategoryTheory.ObjectProperty C)
  [CategoryTheory.LocallySmall.{w', v, u} C],
  (∃ J x, Nonempty (P.FullSubcategory ≌ J)) ↔ CategoryTheory.ObjectProperty.EssentiallySmall.{w, v, u} P