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 $\mathbb{R}$?



yoneda-embedding vs. dual vector space
I don't see the need to try to make vector spaces into categories. I would just say that in each case we have a closed symmetric monoidal category (respectively Vect or Cat), a map f : X ⊗ Y → Z for some objects X, Y, Z (respectively $\langle-,-\rangle$ : V ⊗ V → R and Hom : Cop × C → Set) and we are forming the associated map X → hom(Y,Z) (where hom denotes the internal hom functor). The double dual construction is obtained by setting Y = hom(X,Z) and letting f be the evaluation map; it doesn't depend on anything but X and Z.


That said, there is a great analogy between Vect and Cat, where R and Set play parallel roles: but what corresponds to the construction sending C to Hom(Cop, Set) is the free vector space functor from Set to Vect. The analogy goes something like this. (I am omitting some technical conditions for convenience.)



```
sets                categories
vector spaces       cocomplete categories (and colimit-preserving functors)
additive structure  colimits
free v.s. on S      category of presheaves on C
the ground field    Set
(comm.) algebras    cocomplete closed (symmetric) monoidal categories
A-modules           cocomplete V-enriched categories
etc.

```

I am not claiming there is a way to take an object in one column and get a corresponding object in the other column (although under some circumstances that may be possible): rather that it is fruitful to use the left-hand column as a way of thinking about the right-hand column.


See [this nLab page](http://ncatlab.org/nlab/show/free+cocompletion) for an introduction to these ideas.


