Remark 90.11.3. If $L: \text{Mod}^{fg}_ R \to \text{Mod}_ R$ is an $R$-linear functor, then $L$ preserves finite products and sends the zero module to the zero module, see Homology, Lemma 12.3.7. On the other hand, if a functor $\text{Mod}^{fg}_ R \to \textit{Sets}$ preserves finite products and sends the zero module to a one element set, then it has a unique lift to a $R$-linear functor, see Lemma 90.11.4.

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