The Stacks project

Definition 46.3.1. Let $A$ be a ring. A module-valued functor is a functor $F : \textit{Alg}_ A \to \textit{Ab}$ such that

1. for every object $B$ of $\textit{Alg}_ A$ the group $F(B)$ is endowed with the structure of a $B$-module, and

2. for any morphism $B \to B'$ of $\textit{Alg}_ A$ the map $F(B) \to F(B')$ is $B$-linear.

A morphism of module-valued functors is a transformation of functors $\varphi : F \to G$ such that $F(B) \to G(B)$ is $B$-linear for all $B \in \mathop{\mathrm{Ob}}\nolimits (\textit{Alg}_ A)$.