∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {Φ : CategoryTheory.SmallObject.SuccStruct C}
  {J : Type u} [inst_1 : LinearOrder J] [inst_2 : OrderBot J] [inst_3 : SuccOrder J] [inst_4 : WellFoundedLT J]
  [inst_5 : CategoryTheory.Limits.HasIterationOfShape J C] {j : J} (iter : (i : J) → i < j → Φ.Iteration i)
  {i₁ i₂ : ↑(Set.Iio j)} (f : i₁ ⟶ i₂),
  (CategoryTheory.SmallObject.SuccStruct.Iteration.mkOfLimit.inductiveSystem iter).map f =
    (iter ↑i₁ ⋯).mapObj (iter ↑i₂ ⋯) ⋯ ⋯ ⋯ ⋯