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?