We assume that $f$ is cyclic and we choose a vector $x_0$ in $E$ such that $(x_0, f(x_0), \ldots, f^{n-1}(x_0))$ is a basis of $E$. Let $g \in \mathcal{C}(f)$, an endomorphism that commutes with $f$, and suppose $g(x_0) = \sum_{k=0}^{n-1} \lambda_k f^k(x_0)$.
Show then that $g \in \mathbb{K}[f]$.