Commutative Linear Transformation is G-Module Homomorphism

Theorem
Let $\rho:G\to\operatorname{GL}\left(V\right)$ be a representation, and $f:V\to V$ a linear map. Then:

$\rho(g)\circ f=f\circ \rho(g)$ for all $g\in G\ \Rightarrow\ f:V\to V$ is a $G$-module homomorphism.

Proof
Assume that $\rho(g)\circ f=f\circ \rho(g)$ for all $g\in G$.

Let $v$ be a vector $v\in V$.

Then $\rho(g)(f(v))=f(\rho(g)(v))$.

Using the properties from Equivalence of Representation Definitions there exists $\left(V,\phi\right)$, a $G$-module associated with $\rho$, such that $\phi(g,v)=\rho(g)(v)$.

Applying the last formula:

$\rho(g)(f(v))=\phi(g,f(v))$

and

$f(\phi(g,v))=f(\rho(g)(v))$.

Thus our assumption is equivalent to $f(\phi(g,v))=\phi(g,f(v))$.

Hence by definition of $G$-module homomorphism $f:V\to V$ is a $G$-module homomorphism.