Let Met denote the category of metric spaces and nonexpansive maps. It's well known that if we equip the product of two metric spaces with the L_{\infty} metric (the max of the distances in the two coordinates), we get categorical products in Met; alternatively, if we impose the L_1 metric on the product (the sum of the two coordinate distances), we get a monoidal closed structure, at least if we weaken the usual definition of a metric by allowing metrics to take the value \infty. It's intuitively obvious that the cartesian monoidal structure on Met can't be closed. But I've never (until I wrote one down today!) seen a formal proof of this; does anyone know if it exists anywhere in the literature? My proof is not particularly elegant: it amounts to showing that a particular coequalizer in Met is not preserved by a functor of the form (-) x Y. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]