\begin{definition}[Definition:G-Module Homomorphism]
Let $\left({G, \cdot}\right)$ be a group.
Let $\left({V, \phi}\right)$ and $\left({W, \mu}\right)$ be $G$-modules.
Then a linear transformation $f: V \to W$ is called a '''$G$-module homomorphism''' {{iff}}:
:$\forall g \in G: \forall v \in V: f \left({\phi \left({g, v}\right)}\right) = \mu \left({g, f \left({v}\right)}\right)$
\end{definition}