Let $\phi:\mathcal{F}\to\mathcal{G}$ be a morphism of sheaves of $R$-modules over a topological space $X$.
I'm aware of the fact that $\phi$ is an epimorphism $\Leftrightarrow \phi_x$ is surjective forall $x\in X$.
I'm trying to prove/disprove the following:
$\phi$ is an epimorphism $\Leftrightarrow $ every $x\in X$ has an open neighbourhood $U\subset X$ such that $\phi_U:\mathcal{F}(U)\to\mathcal{G}(U)$ is surjective.
The converse $(\Leftarrow)$ is easy, since if $\phi_U$ is surjective, clearly $\phi_x$ is surjective and by arbitrariness of $x$, $\phi$ is an epimorphism.
For $(\Rightarrow)$, I don't know if it's possible to find such $U$.
Any suggestions? Is this just false?