Definition 89.11.1. Let $L: \text{Mod}^{fg}_ R \to \text{Mod}_ R$, resp. $L: \text{Mod}_ R \to \text{Mod}_ R$ be a functor. We say that $L$ is *$R$-linear* if for every pair of objects $M, N$ of $\text{Mod}^{fg}_ R$, resp. $\text{Mod}_ R$ the map

\[ L : \mathop{\mathrm{Hom}}\nolimits _ R(M, N) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ R(L(M), L(N)) \]

is a map of $R$-modules.

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