A tale of two categories: Inductive groupoids and cross-connections

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