I have been learning about sheaves and am thinking about the following problem. Let $F$ and $G$ be sheaves, say of abelian groups, on a space $X$. The sheaf $Hom(F, G)$ is defined by $Hom(F, G)(U)=Mor(F|_U, G|_U)$. Given a point $p \in X$ and an open set $U$ containing $p$, a morphism $\varphi: F|_U \rightarrow G|_U$ induces a homomorphism on stalks $\phi: F_p \rightarrow G_p$, which is an element of $Hom(F_p, G_p)$. Thus, by the universal property of direct limits, we have a homomorphism from $Hom(F, G)_p$ to $Hom(F_p, G_p)$. However, this is not in general injective or surjective. Why not? An example or a hint leading towards an example would be much appreciated. I have thought about this for some simple sheaves (such as skyscraper sheaves), but it seems to be true in those cases.
I am also interested in a more general answer if there is one, i.e. something category theoretic about Hom and direct limits.