  The importance of accessible categories has been widely recognized; they can
be described as those freely generated in some precise sense by a small set of
objects and, because of that, satisfy many good properties. More specifically
finitely accessible categories can be characterized as: (a) free cocompletions
of small categories under filtered colimits, and (b) categories of flat
presheaves on some small category. The equivalence between (a) and (b) is what
makes the theory so general and fruitful.
  Notions of enriched accessibility have also been considered in the literature
for various bases of enrichment, such as
$\mathbf{Ab},\mathbf{SSet},\mathbf{Cat}$ and $\mathbf{Met}$. The problem in
this context is that the equivalence between (a) and (b) is no longer true in
general. The aim of this paper is then to:
  (1) give sufficient conditions on $\mathcal V$ so that (a) $\Leftrightarrow$
(b) holds;
  (2) give sufficient conditions on $\mathcal V$ so that (a) $\Leftrightarrow $
(b) holds up to Cauchy completion;
  (3) explore some examples not covered by (1) or (2).
