  Cross-connection theory provides the construction of a semigroup from its
ideal structure using small categories. A concordant semigroup is an
idempotent-connected abundant semigroup whose idempotents generate a regular
subsemigroup. We characterize the categories arising from the generalised Green
relations in the concordant semigroup as consistent categories and describe
their interrelationship using cross-connections. Conversely, given a pair of
cross-connected consistent categories, we build a concordant semigroup. We use
this correspondence to prove a category equivalence between the category of
concordant semigroups and the category of cross-connected consistent
categories. In the process, we illustrate how our construction is a
generalisation of Nambooripad's cross-connection analysis of regular
semigroups. We also identify the inductive cancellative category associated
with a pair of cross-connected consistent categories.
