(a) Limits of Cauchy sequences in a (possibly non-symmetric) metric space are shown to be weighted colimits (a notion introduced by Borceux and Kelly, 1975). As a consequence, further insights from enriched category theory are applicable to the theory of metric spaces, thus continuing Lawvere's (1973) approach. Many of the recently proposed definitions of generalized limit turn out to be theorems from enriched category theory. (b) The dual of the space of metrical predicates (`fuzzy subsets') of a metric space is shown to contain the collection ${cal F$ of formal balls (Weihrauch and Schreiber, 1981; Edalat and Heckmann, 1996) as a quasi-metric subspace. Formal balls are related to ordinary closed balls by means of the Isbell conjugation. For an ordinary metric space $X$, the subspace of minimal elements of F is isometric to $X$ by the co-Yoneda embedding.

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Rutten, J.J.M.M. (1997). Weighted colimits and formal balls in generalized metric spaces. Software Engineering [SEN]. CWI.