I am wondering if the following proof to show that a morphism $\phi:F\rightarrow G$ of sheaves is isomorphic iff it is injective and surjective is correct:
Suppose it's injective, then $F$ can be identified with a subsheaf of $G$ and from surjectivity this subsheaf must be the whole $G$, so that $\phi$ is an isomorphism of sections, thus sheaves.
Conversely, an isomorphism must be isomorphism on sections, so in particular injective, and then the image of $F$ as subsheaf of $G$ is just $G$ itself, so surjective.