Duality for powerset coalgebras.

We describe the endofunctor $H$ on the category $CABA$ of complete and atomic boolean algebras and complete boolean homomorphisms such that the category $Alg(H)$ of algebras for $H$ is dually equivalent to the category $Coalg(\mathcal{P})$ of coalgebras for the powerset endofunctor $\mathcal{P}$ on $Set$. As a consequence, we derive Thomason duality from Tarski duality.