  We prove a number of results involving categories enriched over
\textsc{CMet}, the category of complete metric spaces with possibly infinite
distances. The category \textsc{CPMet} of intrinsic complete metric spaces is
locally $\aleph_1$-presentable, closed monoidal, and comonadic over
\textsc{CMet}. We also prove that the category \textsc{CCMet} of convex
complete metric spaces is not closed monoidal and characterize the
isometry-$\aleph_0$-generated objects in \textsc{CMet}, \textsc{CPMet} and
\textsc{CCMet}, answering questions by Di Liberti and Rosick\'{y}. Other
results include the automatic completeness of a colimit of bi-Lipschitz
morphisms of complete metric spaces and a characterization of those pairs
(metric space, unital $C^*$-algebra) that have a tensor product in the
\textsc{CMet}-enriched category of unital $C^*$-algebras.
