  Hilbert modules over a $C^*$-category were first defined by Mitchener, who
also proved that they form a $C^*$-category. An Eilenberg-Watts theorem for
Hilbert modules over $C^*$-algebras was proved by Blecher. We follow a similar
path to prove an Eilenberg-Watts theorem for Hilbert modules over
$C^*$-categories and characterize equivalences of categories of Hilbert modules
as being given by tensoring with imprimitivity bimodules. We employ our results
to prove several equivalences of bicategories of $C^*$-algebras and
$C^*$-categories, and to exhibit a Morita localization of the category of
locally small $C^*$-categories.
