TITLE: Are there other dualities on finite vector spaces besides the canonical one?
QUESTION [8 upvotes]: Let $\text{FinVec}$ denote the category of finite dimensional vector spaces over some field $k$, and let $F:\text{FinVec}\to \text{FinVec}$ be a contravariant functor such that $F^2$ is naturally isomorphic to the identity. Is $F$ naturally isomorphic to the canonical duality functor $V\mapsto V^*=\text{Hom}(V,k)$?
I suspect this question has been asked before on MO, but I couldn't find it.

REPLY [10 votes]: $FinVec$ and its opposite are enriched in finite dimensional $k$-vector spaces. Assume that $F$ is an enriched functor. Then consider the covariant functor:
$$F(-)^*: FinVec \to FinVec$$ 
It is a $Vec$-enriched functor. Finite direct sums are absolute limits and so are preserved by this functor (see here). Since every finite dimensional vector space is isomorphic to a finite direct sum of copies of $k$ it follows that this functor is completely determined by the single vector space $L=F(k)^*$, and there is a natural isomorphism:
$$F(V)^* \cong V \otimes L$$
Thus $F(V) \cong V^* \otimes L^*$ as functors. Your condition that $F^2 \cong id$ implies that $L$ must be one-dimensional (hence isomorphic to $k$) and so $F(-) \cong (-)^*$ is the standard duality functor.