∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {W : CategoryTheory.MorphismProperty C}
  [W.IsStableUnderBaseChange] (J : Type u'') [inst_2 : CategoryTheory.Category.{v'', u''} J]
  [CategoryTheory.Limits.HasPullbacks C], (W.functorCategory J).IsStableUnderBaseChange