What is a monoidal metric space? 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.
 A: Did you follow the thread Simon Willerton started on profunctors between metric spaces, which took us all the way to optimal transport theory?
A: 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.

A: For the general question about how categorical concepts look when applied to metric spaces, one place to look is Lawvere's paper 'Taking categories seriously', section 6 onwards.  
Aside from that, here are a few examples.  
Functor categories became function spaces with the uniform or sup metric.  That is, if $A$ and $B$ are metric spaces construed as enriched categories, then the functor category $B^A$ is the set of distance-decreasing maps $A \to B$ with the sup metric.  (I use "decreasing" in the non-strict sense.)
The (cartesian) product $A \times B$ of two metric spaces --- that is, their product in the category of metric spaces --- has the '$\infty$-metric': 
$$
d((a, b), (a', b')) = \max\{d(a, a'), d(b, b')\}.
$$
The same goes for infinite products --- remembering that $\infty$ is allowed as a distance.  Once you know this, limits in general work in the obvious way.
The coproduct $A + B$ of two metric spaces $A$ and $B$ is their disjoint union, with $d(a, b) = d(b, a) = \infty$ for all $a \in A, b \in B$.  Again, it's crucial here to allow $\infty$ as a distance.  Otherwise, your category of metric spaces will lack lots of limits and colimits.  The coequalizer of two maps $f, g: A \to B$ is $B$ quotiented out by the usual equivalence relation $\sim$ (as in the category of sets), and metrized by
$$
d([b], [b']) = \inf\{ d(y, y'): y \sim b, y' \sim b'\}
$$
where $[b]$ denotes the equivalence class of $b$.  General colimits work similarly.
I mentioned the cartesian product, but there's another kind of product.  Generally, if $\mathbf{V}$ is a monoidal category then any two $\mathbf{V}$-enriched categories, $A$ and $B$, have a tensor product $A \otimes B$.  Its set of objects is the product of the sets of objects of $A$ and $B$.  Its hom-objects are given by 
$$
(A \otimes B)((a, b), (a', b')) = A(a, a') \otimes B(b, b').
$$
This gives us a tensor product of metric spaces.  Given metric spaces $A$ and $B$, the point-set of $A \otimes B$ is the product of the point-sets of $A$ and $B$.  The distance is given by
$$
d((a, b), (a', b')) = d(a, a') + d(b, b').
$$
In other words, it's the '$1$-metric', also known as the taxicab metric, Manhattan metric, etc.  
So, Andrew, when you ask 'What is a monoidal metric space?', you have to say which product you want to be monoidal with respect to.  That is, are you asking about (weak) monoids in $(\mathbf{Met}, \times)$ or in $(\mathbf{Met}, \otimes)$?  
From the tone of your question, I would guess: both.  So here goes.
The answer for cartesian product $\times$ doesn't seem so interesting.  Assuming that your metric spaces satisfy the classical skeletality axiom ($d(a, b) = 0 \Rightarrow a = b$), a monoidal category for $\times$ is a metric space $A$ equipped with a monoid structure on its set of points such that
$$
d(a \cdot b, a' \cdot b') \leq \max\{d(a, a'), d(b, b')\}.
$$ 
I can't think of anything more to say about that.  
The answer for tensor product $\otimes$ seems more interesting.  A monoid in $(\mathbf{Met}, \otimes)$ is a metric space $A$ equipped with a monoid structure on its set of points such that for all $a$, the maps $a\cdot -$ and $- \cdot a$ are distance-decreasing.  For example, if it's a group then this says that left or right translation is always an isometry.  This often happens: consider the underlying additive group of a normed vector space, for instance.
