Lemma 90.11.13. Let $F, G: \mathcal{C} \to \textit{Sets}$ be functors satisfying the hypotheses of Lemma 90.11.8. Let $t : F \to G$ be a morphism of functors. For any $M \in \mathop{\mathrm{Ob}}\nolimits (\text{Mod}^{fg}_ R)$, the map $t_{R[M]}: F(R[M]) \to G(R[M])$ is a map of $R$-modules, where $F(R[M])$ and $G(R[M])$ are given the $R$-module structure from Lemma 90.11.8. In particular, $t_{R[\epsilon ]} : TF \to TG$ is a map of $R$-modules.

**Proof.**
Follows from Lemma 90.11.5.
$\square$

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