Lemma 90.11.7. Let $R$ be an $S$-algebra. Then the functor $\text{Mod}_ R \to S\text{-Alg}/R$ described above preserves finite products.

**Proof.**
This is merely the statement that if $M$ and $N$ are $R$-modules, then the map $R[M \times N] \to R[M] \times _ R R[N]$ is an isomorphism in $S\text{-Alg}/R$.
$\square$

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