I am working on a problem where I'm being asked to find a topological space $X$ that consists of $3$ points, sheaves $\mathscr{F}$ and $\mathscr{G}$ such that a morphism $\varphi$ between $\mathscr{F}$ and $\mathscr{G}$ is surjective on stalks i.e the map $\varphi_x$ is surjective, but the map on sections $\varphi(U) : \mathscr{F}(U) \to \mathscr{G}(U)$ is not surjective.
Of the top of my head I knew that $X = \operatorname{Spec} \mathbb{Z}/60 \mathbb{Z}$ consists of $3$ points namely $X = \{(2), (3), (5)\}$ and given the Zariski topology would be a topological space with $3$ points. Now I would need to figure out how to construct $\mathscr{F}$, $\mathscr{G}$ and $\varphi$ such that the map on stalks is surjective, but the map on sections is not. Is there some general way to think about this construction? In a sense I need to find some open set $U$ for which $\mathscr{G}(U)$ contains points that $\mathscr{F}(U)$ doesn't have a preimage. In the case with $X$ the open sets are $$\emptyset, \{(2)\}, \{(3)\}, \{(5)\}, \{(2), (3)\}, \{(2), (5)\}, \{(3), (5)\}, X$$ if I didn't miss any. Any hints on how to construct suitable sheafs for this?