Let $V, W, U$ be finite dimensional representations of a lie algebra $\mathfrak{g}$. Show that $\hom(V \otimes W, U) \cong \hom (V, U \otimes W^*)$.
I think I have to use the enveloping algebra of the lie algebra here but I can't find a natural isomorphism. I have tried in some simple finite cases but they aren't very illuminating. Any guidance would be appreciated, thanks.