Let A be a finite algebra with a majority term. We characterize those algebras categorically equivalent to A. The description is in terms of a derived structure with universe consisting of all subalgebras of A × A, and with operations of composition, converse and intersection.

The main theorem is used to get a different sort of characterization of categorical equivalence for algebras generating an arithmetical variety. We also consider clones of co-height at most two. In addition, we provide new proofs of several characteriza- tions in the literature, including quasi-primal, lattice-primal and congruence-primal algebras.