The Stacks project

Lemma 18.26.1. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a presheaf of rings. Let $\mathcal{F}$, $\mathcal{G}$ be presheaves of $\mathcal{O}$-modules. Then $\mathcal{F}^\# \otimes _{\mathcal{O}^\# } \mathcal{G}^\# $ is equal to $(\mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G})^\# $.

Proof. Omitted. Hint: use the characterization of tensor product in terms of bilinear maps below and use the universal property of sheafification. $\square$

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