∀ {X Y : AddCommMonCat} (f : X ⟶ Y),
  CategoryTheory.CategoryStruct.comp
      ((CategoryTheory.coyoneda.obj (Opposite.op (AddCommMonCat.of (ULift.{u_1, 0} ℕ)))).map f)
      ((fun M => (CategoryTheory.ConcreteCategory.homEquiv.trans (uliftMultiplesHom ↑M).symm).toIso) Y).hom =
    CategoryTheory.CategoryStruct.comp
      ((fun M => (CategoryTheory.ConcreteCategory.homEquiv.trans (uliftMultiplesHom ↑M).symm).toIso) X).hom
      ((CategoryTheory.forget AddCommMonCat).map f)