Can we take stalks of morphisms of pre-sheaves in general categories?

Question

Suppose we have a topological space $X$ and a pre-sheaf $\mathcal F$ over it. My question would be: Is it possible, when $\mathcal F$ is merely a pre-sheaf in an arbitrary category $\mathcal C$ (ie. a functor $\operatorname{Ouv}(X) \to \mathcal C$, $\operatorname{Ouv}(X)$ being the category of open sets of $X$ with morphisms all inclusions $V \subseteq U$), to define, whenever we also have another pre-sheaf $\mathcal G$ and a morphism of pre-sheaves $\phi: \mathcal F \to \mathcal G$, the morphism $\phi_x$ on stalks? Note that taking stalks works in general, and defining the morphism $\phi_x$ works whenever we have a category where we can just explicitly define the stalk as the object of equivalence classes with the eq. rel. being equality on small sets. But I would like a purely categorical way of doing this, and I'm stuck at finding one, whence I'd appreciate any aid.