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When one talks about the category $V_K$ of vector spaces over a field $K$ and considers the dual functor $D$ which maps a vector space $V$ to its dual $V^{*}$ I believe to have in mind something like a labeled category, the labels letting me know which object is the dual of another object. (Or can I see this by carefully looking at the morphisms?)

What I want to know:

Is there - analogously to graphs - a distinction between labelled and unlabeled categories?

Side remark: I see something like a predominance of unlabeled graphs over labeled ones, the former being the more "genuine" graphs (as abstract structures). What's the situation in category theory?

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    @Hans: the forgetful functor from $k$-vector spaces to sets is represented by $k$. There is no sense in which you are not allowed to use this functor to make sense of vector spaces.2011-03-11

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$\text{Vect}_k$ is enriched over itself: for any two vector spaces $V, W$, the set $\text{Hom}_k(V, W)$ naturally acquires the structure of a vector space (since linear combinations of linear operators are linear operators), hence defines a functor $\text{Vect}_k \times \text{Vect}_k \to \text{Vect}_k$ contravariant in the first argument.

In particular, for every $V$ the set $\text{Hom}_k(V, k) \equiv V^{\ast}$ naturally acquires the structure of a vector space, and this is the abstract origin of the dual vector space functor $\text{Vect}_k \to \text{Vect}_k$.

By introducing enough extra data, there is no need for "labelings" (and I don't understand what this means anyway). In particular, if you aren't comfortable with singling out $k$ as a $1$-dimensional vector space, you can introduce the monoidal structure on $\text{Vect}_k$, in which the identity object $k$ is singled out as part of the extra data.

Side remark: I do not understand your side remark. There are many categories of graphs, and people are interested in both labeled and unlabeled graphs. What does "genuine" mean?

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    @Hans: I don't know what "standard" means. I still don't know what you mean by "labeling." Introducing extra data is the same in category theory as it is in set theory. My advice to you is the same as always: instead of asking unfocused questions, why don't you _learn and do some mathematics_?2011-03-11