Direct product/Eigenvectors/Exercise

Let $V_1,\dots ,V_n$ denote vector spaces over a field $K$, and let

${\phi }_i:V_i⟶V_i$

denote linear mappings. Let $a\in K$ be an eigenvalue of ${\phi }_k$ for a determined $k$. Show that $a$ is also an eigenvalue of the product mapping

${\phi }_1\times \cdots \times {\phi }_n:V_1\times \cdots \times V_n⟶V_1\times \cdots \times V_n.$

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