Metric exponential candidate ($[X,Y]$)

For bounded metric spaces $X,Y$, let $[X,Y]$ be the set of non-expansive maps and defineWhenever $\bar{d}$ satisfies the triangle inequality, it is a metric and makes $[X,-]$ right adjoint to $-\times X$ in ${\mathbf{Met}}_b$.

A metric space $(X,d)$ is interpolating if, for every $x,y\in X$ and every pair of nonnegative real numbers $r,s$ with $d(x,y)=r+s$, there is $z\in X$ with $d(x,z)=r$ and $d(z,y)=s$. Thus each distance can be split at any prescribed fraction. The nonnegativity requirement is essential.

A convex set in a normed vector space is interpolating: when $r+s>0$, take $z=x+\frac{r}{r+s}(y-x)$; when $r+s=0$, take $z=x=y$. A two-point metric space with the discrete metric is not interpolating, because a distance of one has no midpoint at distance $1/2$ from both endpoints. Interpolation makes the metric exponential candidate satisfy the triangle inequality, so every bounded interpolating metric space is an exponentiable object of the category of bounded metric spaces and non-expansive maps.