TITLE: Stalks of the tensor product presheaf of two sheaves
QUESTION [19 upvotes]: Let $(X, \mathscr{O})$  be a ringed space and $\mathscr{F}, \mathscr{G}$  be sheaves of $\mathscr{O}$-modules on $X$.
Define $\mathscr{H}(U) = \mathscr{F}(U) \otimes_{\mathscr{O}(U)} \mathscr{G}(U)$. I am stuck trying to prove that $\mathscr{H}_p \cong \mathscr{F}_p \otimes_{\mathscr{O}_p} \mathscr{G}_p$ as $\mathscr{O}_p$-modules.
I know that if $X$  is a topological space, $\mathscr{F}, \mathscr{G}$ are presheaves of abelian groups on $X$ and $\mathscr{H}(U) = \mathscr{F}(U) \otimes_{\mathbb{Z}} \mathscr{G}(U)$ then $\mathscr{H}_p \cong \mathscr{F}_p \otimes_{\mathbb{Z}} \mathscr{G}_p$. This is just a consequence of the fact that tensor products commute with direct limits. But I don't know how to deal with the case when the base ring is changing

REPLY [6 votes]: Alternatively you can prove that the tensor product commutes with colimits not only in both factors, but also over the base ring. This follows easily from the adjunction.