Morphism Of Invertible Sheaves

[Tsakanikas Nickos]

Let \( \left( X, \mathcal{O}_{X} \right) \) be a scheme and let \( \displaystyle \phi \ \colon \mathcal{L} \longrightarrow \mathcal{M} \) be a morphism of invertible sheaves on \( X \). Show that if \( \phi \) is surjective, then \( \phi \) is actually an isomorphism. Does the same hold if instead we suppose that \( \phi \) is injective?