TITLE: yoneda-embedding vs. dual vector space
QUESTION [11 upvotes]: I've made the following observation: let V be a vector space over $\mathbb{R}$ with a inner product $\langle , \rangle$. then there is a "natural contravariant" injective map $V \to \hom(V,\mathbb{R})$. if we apply this twice, we get a "natural covariant" injective map $V \to \hom(\hom(V,\mathbb{R}),\mathbb{R}), v \mapsto (\phi \mapsto \phi(v))$. but the same things happen in category theory: let $C$ be a category (which I assume to be locally-small), then $C \to \hom(C,Set), x \mapsto \hom(x,-)$ is a natural contravariant fully-faithful functor and applying this twice yields (up to natural isomorphism) the natural covariant functor $C \to \hom(\hom(C,Set),Set), x \mapsto (F \mapsto F(x))$ (yoneda-lemma).
so how can we unify these two phenomena? perhaps we can make $V$ to a enriched category over $\mathbb{R}$ and hope that the yoneda-lemma for symmetric closed monoidal categories is the common generalization? what is the right structure on $\mathbb{R}$?

REPLY [4 votes]: There are many different things that Martin may have had in mind in asking this question, but the obvious answer requires less category theory and not more.
The basic operation is   ${-}\to S$   for some fixed "dualising" object in some cartesian closed (or more generally monoidal closed) category.
This operation is part of a monad, of which Martin's natural map is the unit.