ct.category theory - 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 : C^op × 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(C^op, 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 for an introduction to these ideas.