Let $U,V$ and $W$ be finite dimensional vector spaces, and define $B$ to be the vector space of all bilinear maps $V \times W \to \mathbb{R}$. Given a bilinear map $\alpha : V \times W \rightarrow U$, define $\tilde{\alpha}: B^* \rightarrow U^{**}$ by $\alpha(\psi)(\sigma) = \psi (\sigma \circ \alpha)$. Define a map $\pi : V \times W \rightarrow B^*$ by $\pi(v,w) (f:V \times W \rightarrow \mathbb{R}) = f(v,w).$
$\mathbf{CORRECTION:}$ $B$ should be the space of bilinear maps $V \times W \to \mathbb{R}$, not $V \times W \to U$ as previously stated.
In order to show that $B^*$ satifies the universal property of the tensor product, I have to show that given a map $\alpha : V \times W \rightarrow U$, then there is a unique $\tilde{\alpha} : B^* \rightarrow U^{**}$ such that $\Theta \circ \tilde{\alpha} \circ \pi = \alpha$, where $\Theta:U^{**} \to U$ is the canonical isomorphism.
It is quite clear that $\tilde{\alpha}$ defined above satisfies this property, but I am having trouble proving uniqueness. I would like to show that given $f:B^* \to U^{**}$ such that $\Theta\circ f \circ \pi = \alpha$, then $f= \tilde{\alpha}$, however I am getting nowhere. Any help would be appreciated, thank you.