  We obtain an ESN theorem for a very general class of biunary semigroups with
idempotent-valued domain and range operations, representing them in terms of
small categories equipped with a suitable biaction of the identities on the
category. Our results generalise the recent work of Fitzgerald and Kinyon
connecting localisable semigroups to transcription categories, as well as that
of Lawson linking Ehresmann semigroups to categories with Ehresmann biaction.
In contrast to most approaches to ESN theorems, we do not require the
categories to be ordered or for their sets of identities to possess any
particular structure. Throughout, the biunary semigroups are represented using
categories rather than generalised categories of any kind, and we obtain
category isomorphisms between the clesses of semigroups and their associated
enriched categories, rather than category equivalences. Our results cover the
class of DRC-semigroups considered by Jones and Shoufeng Wang, but they also
cover cases where not both congruence conditions hold, including examples such
as the semigroup of binary relations on a set under demonic composition
equipped with domain and range operations.
