The Stacks project

Lemma 90.11.5. Let $L_1, L_2: \text{Mod}^{fg}_ R \to \textit{Sets}$ be functors that take $0$ to a one element set and preserve finite products. Let $t : L_1 \to L_2$ be a morphism of functors. Then $t$ induces a morphism $\widetilde{t} : \widetilde{L}_1 \to \widetilde{L}_2$ between the functors guaranteed by Lemma 90.11.4, which is given simply by $\widetilde{t}_ M = t_ M: \widetilde{L}_1(M) \to \widetilde{L}_2(M)$ for each $M \in \mathop{\mathrm{Ob}}\nolimits (\text{Mod}^{fg}_ R)$. In other words, $t_ M: \widetilde{L}_1(M) \to \widetilde{L}_2(M)$ is a map of $R$-modules.

Proof. Omitted. $\square$

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