Two algebraic structures A and B are called categorically equivalent if there is a functor from
the variety generated by A to the variety generated by B, carrying A to B, that is an
equivalence of the varieties when viewed as categories. We characterize those algebras
categorically equivalent to A when A is an algebra whose set of term operations is as large
as possible subject to constraints placed on it by the subalgebra or congruence lattice of A,
or the automorphism group of A.