  We study presheaves on semicategories enriched in a quantaloid: this gives
rise to the notion of regular presheaf. A semicategory is regular when its
representable presheaves are regular, and its regular presheaves then
constitute an essential (co)localization of the category of all of its
presheaves. The notion of regular semidistributor allows to establish the
Morita equivalence of regular semicategories. Continuous orders and Omega-sets
provide examples.
