If we consider metric spaces to be categories enriched over $\mathbb R\_{\geq 0}$, the object corresponding to presheaves should be lipschitz-continuous functions $\operatorname{Lip^ 1}(M, \mathbb R\_{\geq 0})$. Now there should be an obvious metric on this set; making the Yoneda map $$x\mapsto \operatorname d(-,x)$$
an isometric embedding. **What is this metric?**



Reference Request(Enriched Categories): Metric on Lipschitz Continuous Functions
It is the usual sup metric. See section 2 of Lawvere's original [article](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html).


