TITLE: What is a monoidal metric space?
QUESTION [14 upvotes]: At time of writing, the highest rated answer to my question What is a metric space? is Tom Leinster's account of Lawvere's description of a metric space as an enriched category.  This prompted my question on terminology in category theory.  That question was focussed on terms in category theory that were previously in use elsewhere.  It now occurs to me that there's an obvious question in the other direction: 
Given that metric spaces are enriched categories, what do the standard categorical things look like?
Obvious ones are adjoint functors (which might help me get a picture of what adjoint functors really are, answering this question), monoidal structures (and symmetric monoidal structures), products and coproducts (more generally, limits and colimits), but I'm sure that there are many more "out there" and I don't want to limit the answers.
To forestall Urs Schrieber's likely first comment, I intend to stick all of this on an n-lab page sometime soon as, if my intuition is right, I think it might be a neat case study that can help topologists like me get a picture of how categories (and enriched categories) can behave.

REPLY [5 votes]: Here are a couple of answers to your wider question.
1) Tom Leinster defined the notion of Euler characteristic for a finite categories, generalizing things like cardinality of sets, Euler characteristic of posets and Euler characteristic of finite groups.  This can be generalized to enriched categories and specialized to metric spaces, giving rise to an (occasionally undefined) invariant of metric spaces called the magnitude.  (The names cardinality and Euler characteristic were deemed to be too confusing.)  Interestingly, this was discovered in the nineties by some ecologists interested in measuring biodiversity.  See our paper [http://arxiv.org/abs/0908.1582](On the asymptotic magnitude of subsets of Euclidean space) for more details.
2) Given an endofunctor F:*C*->C of a category enriched over V there are two ways of taking the 'trace' of F that I know of, both leading to an object of V.  One is the end $\int_c C(c,F(c))$ and the other is the coend $\int^c C(c,F(c))$.  In the context of metric spaces this means that for a distance non-increasing function f:*X*->X the two traces are supxd(x,f(x))
and infxd(x,f(x)) - which can be thought of as the furthest distance that f moves points and the least distance that f moves points.