Coalgebras for the powerset functor and Thomason duality

Let $CABA$ be the category of complete and atomic boolean algebras and complete boolean homomorphisms, and let $CSL$ be the category of complete meet-semilattices and complete meet-homomorphisms. We show that the forgetful functor from $CABA$ to $CSL$ has a left adjoint. This allows us to describe an endofunctor $H$ on $CABA$ 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.