Tensor product/Linear mappings/Composition/Exercise

Let ${}K$ be a field, and let ${}U_{1},\ldots ,U_{n},V_{1},\ldots ,V_{n},W_{1},\ldots ,W_{n}$ denote vector spaces over ${}K$ . Let

\psi _{i}\colon U_{i}\longrightarrow V_{i}

and

\varphi _{i}\colon V_{i}\longrightarrow W_{i}

be ${}K$ -linear mappings. Show

{}(\varphi _{1}\circ \psi _{1})\otimes _{}\cdots \otimes _{}(\varphi _{n}\circ \psi _{n})=(\varphi _{1}\otimes _{}\cdots \otimes _{}\varphi _{n})\circ (\psi _{1}\otimes _{}\cdots \otimes _{}\psi _{n})\,.