For every set Z, there is a self-adjoint contravariant functor Q=ENS(--,Z), with unit/counit h:Id-->Q^op Q given by (h_X)(x)(f)=f(x). Let Q-alg denote the category of algebras for the monad induced by this self-adjunction. If Z is not empty or a singleton, then the comparison functor ENS^op-->Q-alg is an equivalence by results of M. Sobral. If Z has two members, then Q-alg is isomorphic to CaBool, the category of complete atomic Boolean algebras. What is known about Q-alg if Z has more than two members (beyond the fact that Q-alg and CaBool are equivalent)?