  We give a categorial definition separating cylindric-like algebras from
polyadic-like ones. Viewing the neat reduct operator as a functor, we show that
it does not have a right adjoint in the former case, but it is strongly
invertible in the second case. Several new results on amalgamation, and non
finite axiomatizability are presented for both paradigms. A hitherto categorial
equivalence is also given between relation algebras with quasi-projections and
Nemeti's directed cylindric algebras for any dimension.
