Isomorphism of the space of mixed (k, l+1) - tensors

Question

Let $V$ be a finite-dimensional vector space. There is a natural isomorphism between $T^{k}_{l+1}(V)$ and the space of multilinear maps $$V^{*l}\times V^{k} \rightarrow V$$ I have found an homomorphism $\phi$ from $\{f:V^{*l}\times V^{k} \rightarrow V| \text{f is multilinear}\}$ to $T^{k}_{l+1}(V)$. Say $(\phi A)(\omega^{1},...,\omega^{l+1},X_{1},...,X_{k}) = \omega_{1}(A(\omega^{2},...,\omega^{l+1},X_{1},...,X_{k}))$. How can I prove that $\phi$ is an natural isomorphism?