The Stacks project

Remark 103.10.6. Let $\mathcal{X}$ be an algebraic stack. Given two quasi-coherent $\mathcal{O}_\mathcal {X}$-modules $\mathcal{F}$ and $\mathcal{G}$ the tensor product module $\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}$ is quasi-coherent, see Sheaves on Stacks, Lemma 96.15.1 part (5). Similarly, given two locally quasi-coherent modules with the flat base change property, their tensor product has the same property, see Proposition 103.8.1. Thus the inclusion functors

are functors of symmetric monoidal categories. What is more interesting is that the functor

is a functor of symmetric monoidal categories as well. Namely, given $\mathcal{F}$ and $\mathcal{G}$ in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ we obtain

where the south-west arrow comes from the universal property of the north-west arrow (and the fact already mentioned that the object in the upper left corner is quasi-coherent). If we restrict this diagram to $U_{\acute{e}tale}$ for $U \to \mathcal{X}$ flat, then all three arrows become isomorphisms (see Lemmas 103.10.1 and 103.10.2 and Definition 103.9.1). Hence $Q(\mathcal{F}) \otimes _{\mathcal{O}_\mathcal {X}} Q(\mathcal{G}) \to Q(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G})$ is an isomorphism, see for example Lemma 103.4.2.