If I have a map of sheaves $\varphi : \mathscr{F}\rightarrow \mathscr{G}$ on some topological space $X$, where $\mathscr{F}$ is flasque (i.e. all restriction maps are surjective), is it true that the map is in fact fully determined by the map $\Gamma(X,\mathscr{F}) \rightarrow \Gamma(X,\mathscr{G})$?
My thinking is that for an open set $U\subset X$ and a section $s \in \Gamma(U,\mathscr{F})$ we can extend to a global section $\tilde{s} \in \Gamma(X,\mathscr{F})$ since $\mathscr{F}$ is flasque, and then restrict its image under the map $\varphi(X)$ to the open set $U$ to get the image of $s$ under $\varphi(U)$. This should be the case since maps of sheaves commute with restriction maps by definition. Indeed, is this true for a $\varphi$ a map of presheaves?