It is very well known that if $\alpha : \mathscr{F} \to \mathscr{G}$ is a morphism of sheaves, then it induces homomorphisms on the stalks. I have been wondering for a while if given a collection of homomorphisms between the stalks of two sheaves and some suitable patching condition, can we construct a homomorphism of sheaves?

I have been thinking about this problem a little bit lately, but I have not been able to come up with a "canonical" solution. In other words I have found ways of producing sheaf morphisms from stalk morphisms but they have always felt very unnatural.

Let me phrase the question a little bit more formally. Let $X$ be a topological space and $\mathscr{F}, \mathscr{G}$ sheaves on $X$. Let $\alpha_p : \mathscr{F}_p \to \mathscr{G}_p$ be a homomorphism for each $p \in X$. Is there a suitable condition on the $\alpha_p$s such that there exists a sheaf morphism $\alpha : \mathscr{F} \to \mathscr{G}$ whose induced maps on the stalks are exactly the $\alpha_p$s?