A generalization of the Ehresmann–Schein–Nambooripad theorem to two-sided Ehresmann semigroupoids

In this paper, we introduce the notion of two-sided Ehresmann semigroupoids and show that they are in correspondence with a specific class of categories, which we call local biordered Ehresmann categories. This correspondence provides a unified generalization of the Ehresmann–Schein–Nambooripad Theorem for both inverse semigroupoids and Ehresmann semigroups. In particular, two-sided restriction semigroupoids form a distinguished subclass of two-sided Ehresmann semigroupoids, and for this case we describe the associated class of categories, extending earlier results for restriction semigroups.