  This paper is part of a project that aims to give a homotopy cousin of
Kelly's treatment of enriched category theory. After introducing unital
co-Segal M-categories, we establish the unital version of a previous theorem
that was proven for the nonunital ones; but this was done under strong
hypothesis. We've removed here these assumptions and try to keep the hypothesis
on M as minimal as possible. Our main result provides a sort of Bousfield
localization of a model category that is not known to be left proper. In this
model structure, every co-Segal M-category is `canonically' equivalent to a
strict M-category with the same set of objects. We revisit some constructions
of classical enriched category theory to set up the necessary material for our
future applications.
