"Let $(X, \mathcal{O})$ be a ringed space. A sheaf of modules $\mathcal{F}$ on $X$ is finitely generated if for all $a \in X$ there exists a neighbourhood $U$ of $a$, an integer $n$ and a surjective morphism $\phi: \mathcal{O}_U^n \to \mathcal{F}_U$."
Here $\mathcal{O}_U$ and $\mathcal{F}_U$ denote the restricted sheaves.
One would like to conclude the following: for any point $a \in X$ there exists a neighbourhood $U$ of $a$ such that for all $V \subseteq U$ the morphism on sections $\phi(V): \mathcal{O}^n(V) \to \mathcal{F}(V)$ is surjective. However, it is well known that a surjective morphism need not be surjective on sections.
My question is: is this claim true and, if yes, how would one go about proving it? Thanks a lot.
