Upon reading the wikipedia article for natural transformations of functors, I stumbled across the section on the dual of a vector space. This is not a question about why the transformation from the identity functor to the dual functor is not natural, rather a worry about how this transformation is defined, set-theoretically speaking.
Denote by $\mathbf{Vec}$ the category of all finite-dimensional vector spaces over some fixed field $k$ with linear maps as morphisms. Then we have a functor $*\colon \mathbf{Vec}^{\mathrm{op}} \to \mathbf{Vec}$ sending a space $V$ to its dual space $V^*$, and a linear map $L$ to its dual map $L^*$. A transformation from $\text{Id}_{\mathbf{Vec}}$ to $*$ requires to choose an isomorphism from $V$ to $V^*$ for every vector space $V$. (This comes down to choosing a nondegenerate bilinear form $V\times V \to k$ for every $V$.)
My worry is that this definition requires 'too many' choices. For any set of vector spaces $\{V_\alpha\}_\alpha$, the axiom of choice would imply that we can make a choice of isomorphism $V_\alpha \to V_\alpha^*$ for all $\alpha$ simultaneously. But of course, the collection of all vector spaces does not form a set, but rather a proper class. As stated, I'm not convinced we can properly define this transformation.
Is there any non-problematic way of defining or talking about such a transformation, then?