The Stacks project

Remark 56.4.2. With $A$, $B$, $F$, and $F'$ as in Lemma 56.4.1. Observe that the tensor product of two finitely presented modules is finitely presented, see Algebra, Lemma 10.12.14. Thus we may endow $\text{Mod}^{fp}_ A$, $\text{Mod}^{fp}_ B$, $\text{Mod}_ A$, and $\text{Mod}_ B$ with the usual monoidal structure given by tensor products of modules. In this case, if $F$ is a functor of monoidal categories, so is $F'$. This follows immediately from the fact that tensor products of modules commutes with filtered colimits.

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