Following up on Qiaochu's query, one way of distinguishing a finite-dimensional V $V$ from an infinite one is that there exists a space W $W$ together with maps e: $e: W \otimes V \to kk$, f: $f: k \to V \otimes W W$ making the usual triangular equations hold(I hope it is clear what those equations are; cf. categorical adjunctions). The data (W, $(W, e, f) f)$ is uniquely determined up to canonical isomorphism, namely W $W$ is canonically isomorphic to the dual of V; $V$; the e $e$ is of course the evaluation pairing. (While it is hard to write down an explicit formula for f: $f: k \to V \otimes V^* V^*$ without referring to a basis, it is nevertheless independent of basis: is the same map no matter which basis you pick, and thus canonical.) By swapping V $V$ and W $W$ using the symmetry of the tensor, there are maps $V \otimes W \to kk$, $k \to W \otimes V V$ which exhibit V $V$ as the dual of W, $W$, hence V $V$ is canonically isomorphic to the dual of its dual.
Just to be a tiny bit more explicit, the inverse to the double dual embedding $V \to V^{**} V^{**}$ would be given by
V^{**}$$V^{\ast\ast} \to V \otimes V^* \otimes V^{**V^{\ast\ast} \to V V$$
where the description of the maps uses the data above.
