I was hoping to use the fact that CABAs are powersets with extra structure on the morphisms to find an endofunctor $F:\text{Set}\to\text{Set}$ with $\text{Set}^{op}\simeq\text{Coalg}F$. I started by seeing that $\text{Coalg}\bar{2}\simeq ( \text{Set}/2)$, where $2$ is the cardinal of the same cardinality, and $\bar{2}$ is the constant functor on said cardinal. Coalgebras on this are just atomic lattices with the usual order-preserving morphisms, right? So we should just be able to build up on $\bar{2}$ to accommodate the extra requirements on these coalgebras that turn them into complete atomic boolean algebras. This is where I am stuck. Unfortunately Google hasn't helped me much, and the nLab page doesn't mention coalgebras of endofunctors.
Also, references on where I can read more about working with algebras and coalgebras of endofunctors are welcome. Thanks!