The Stacks project

Proof. Let $I$ be a preordered set and let $\{ \mathcal{G}_ i\} $ be a system over $I$. Set $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{G}_ i$. Recall that $\mathcal{G}$ is the sheaf associated to the presheaf $\mathcal{G}' : U \mapsto \mathop{\mathrm{colim}}\nolimits _ i \mathcal{G}_ i(U)$, see Sheaves, Section 6.29. By Lemma 17.16.2 the tensor product $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$ is the sheafification of the presheaf

where the equality sign is Algebra, Lemma 10.12.9. Hence the lemma follows from the description of colimits in $\textit{Mod}(\mathcal{O}_ X)$, see Lemma 17.3.2. $\square$