If "f" is quasi-compact and "L" is an invertible sheaf on "X", then "L" is "f"-ample or "f"-very ample if and only if its pullback "L"′ is "f"′-ample or "f"′-very ample, respectively. However, it is not true that "f" is projective if and only if "f"′ is projective. It is not even true that if "f" is proper and "f"′ is projective, then "f" is quasi-projective, because it is possible to have an "f"′-ample sheaf on "X"′ which does not descend to "X".