  Generalising Nachbin's theory of "topology and order", in this paper we
continue the study of quantale-enriched categories equipped with a compact
Hausdorff topology. We compare these $\mathcal{V}$-categorical compact
Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that
the presence of a compact Hausdorff topology guarantees Cauchy completeness and
(suitably defined) codirected completeness of the underlying quantale enriched
category.
