  A groupoid is a small category in which all morphisms are isomorphisms. An
inductive groupoid is a specialised groupoid whose object set is a regular
biordered set and the morphisms admit a partial order. A normal category is a
specialised small category whose object set is a strict preorder and the
morphisms admit a factorisation property. A pair of `related' normal categories
constitutes a cross-connection. Both inductive groupoids and cross-connections
were identified by Nambooripad \cite{mem,cross} as categorical models of
regular semigroups. We explore the inter-relationship between these seemingly
different categorical structures and prove a direct category equivalence
between the category of inductive groupoids and the category of
cross-connections.
