Let $V$, $W$ be finite dimensional vector spaces (over a characteristic zero field $\mathbb{K}$) and $\lambda=(\lambda_1, \cdots, \lambda_n)$ a partition of an integer $m$. Let $L_{\lambda}V$ and $L_{\lambda}W$ be the Schur functor associated to $V$ and $W$. Consider the module $\Lambda^m(V \otimes W)$. How can I prove that there is a natural isomorphism
$$ \Lambda^m(V \otimes W) = \bigoplus_{|\lambda|=m} L_{\lambda}V \otimes L_{\lambda'}W \,\,?$$