  There are two major structure theorems for an arbitrary regular semigroup
using categories, both due to Nambooripad. The first construction using
inductive groupoids departs from the biordered set structure of a given regular
semigroup. This approach belongs to the realm of the celebrated
Ehresmann--Schein--Nambooripad Theorem and its subsequent generalisations. The
second construction is a generalisation of Grillet's work on cross-connected
partially ordered sets, arising from the principal ideals of the given
semigroup. In this article, we establish a direct equivalence between these two
seemingly different constructions. We show how the cross-connection
representation of a regular semigroup may be constructed directly from the
inductive groupoid of the semigroup, and vice versa.
