Boolean algebras are an important kind of poset. The power set functor defines an (contravariant) equivalence \\(P: \text{Set}^{\text{op}} \to \text{Bool}\\): any function \\(f: X \to Y\\) corresponds to the *preimage* map \\(f*: PY \to PX\\), which is a monotone map, and also a boolean algebra homomorphism, meaning it preserves meets and joins (the very important fact that preimage preserves set operations).

Even more interesting, this preimage has a left and right adjoint! **Puzzle CW**: What are they?

Even more interesting, this preimage has a left and right adjoint! **Puzzle CW**: What are they?