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?