Isomorphism between Hom and tensor product I am looking for an explicit isomorphism $Hom(V,V^*)\rightarrow V^*\otimes V^*$ where $V$ is a vector space.
I thought of:
$\phi\rightarrow ((u,v)\rightarrow \phi(u)(v))$
But I'm not sure this works.
Does anyone have a suggestion?

I normally think of it the other way around, that is,
$$
V^{\star} \otimes W \to \hom(V, W),
\qquad
\varphi \otimes w \mapsto (v \mapsto \varphi(v) w).
$$

PS I am assuming $V, W$ to be finite-dimensional, see the comments below. 
