  This paper has two objectives. The first is to develop the theory of
bicategories enriched in a monoidal bicategory -- categorifying the classical
theory of categories enriched in a monoidal category -- up to a description of
the free cocompletion of an enriched bicategory under a class of weighted
bicolimits. The second objective is to describe a universal property of the
process assigning to a monoidal category V the equipment of V-enriched
categories, functors, transformations, and modules; we do so by considering,
more generally, the assignation sending an equipment C to the equipment of
C-enriched categories, functors, transformations, and modules, and exhibiting
this as the free cocompletion of a certain kind of enriched bicategory under a
certain class of weighted bicolimits.
