The Stacks project

Definition 10.8.6. Let $(M_ i, \mu _{ij})$, $(N_ i, \nu _{ij})$ be systems of $R$-modules over the same preordered set $I$. A homomorphism of systems $\Phi$ from $(M_ i, \mu _{ij})$ to $(N_ i, \nu _{ij})$ is by definition a family of $R$-module homomorphisms $\phi _ i : M_ i \to N_ i$ such that $\phi _ j \circ \mu _{ij} = \nu _{ij} \circ \phi _ i$ for all $i \leq j$.