Hilbert modules over $C^{\ast }$-categories

Hilbert modules over a $C^{\ast }$-category were first defined by Mitchener, who also proved that they form a $C^{\ast }$-category. An Eilenberg-Watts theorem for Hilbert modules over $C^{\ast }$-algebras was proved by Blecher. We follow a similar path to prove an Eilenberg-Watts theorem for Hilbert modules over $C^{\ast }$-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 $C^{\ast }$-algebras and $C^{\ast }$-categories, and to exhibit a Morita localization of the category of locally small $C^{\ast }$-categories.