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.



Are there other dualities on finite vector spaces besides the canonical one?
$FinVec$ 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](https://ncatlab.org/nlab/show/Ab-enriched+category)). 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. 


