TITLE: Stalks of sheaf-hom
QUESTION [14 upvotes]: Let $F$ and $G$ be sheaves on $X$. Under what conditions is the natural map from the stalk at $p$ of $\mathcal{H}\kern{-1pt}\mathit{om}(F,G)$ to $\mathrm{Hom}(F_p, G_p)$ an isomorphism?

REPLY [6 votes]: The result in Hartshorne if I recall correctly only really uses the fact that affine locally a coherent sheaf on a scheme has a locally free resolution by finite rank projectives and that one can compute stalks affine locally. In particular, as pointed out by David in the comments we only ready need the first two steps so that one can use the exactness properties of Hom/SheafHom.
So the right condition on F as in the comments is that it be finitely presented.