The Stacks project

Definition 22.24.1. Let $R$ be a ring. An $R$-linear category $\mathcal{A}$ is a category where every morphism set is given the structure of an $R$-module and where for $x, y, z \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ composition law

is $R$-bilinear.

Definition 22.24.2. Let $R$ be a ring. A functor of $R$-linear categories, or an $R$-linear functor is a functor $F : \mathcal{A} \to \mathcal{B}$ where for all objects $x, y$ of $\mathcal{A}$ the map $F : \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(F(x), F(y))$ is a homomorphism of $R$-modules.