∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} D]
  (J : Type w) [inst_2 : LinearOrder J] [inst_3 : SuccOrder J] [inst_4 : OrderBot J] [inst_5 : WellFoundedLT J]
  (G : CategoryTheory.Functor C D) [CategoryTheory.Limits.PreservesWellOrderContinuousOfShape J G] {X Y : C} (f : X ⟶ Y)
  {P : CategoryTheory.MorphismProperty D} [CategoryTheory.Limits.PreservesColimitsOfShape J G],
  (P.inverseImage G).transfiniteCompositionsOfShape J f → P.transfiniteCompositionsOfShape J (G.map f)