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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?

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It is the usual sup metric. See section 2 of Lawvere's original article.

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