Adjoint Functor Theorem For Posets

The adjoint functor theorem for Posets is a specialized form of the Adjoint Functor Theorem that says that every Monotone, Meet preserving map $f:X\to Y$ from an Inflattice has a Left Adjoint $f_{!}:Y\to X$.

The proof is delightfully simple: every Inflattice is a Complete Category, and $f$ preserves all Limits as a Functor, so all that remains is to check the Solution Set Condition. However, the solution set condition trivially holds in a poset; in fact, the whole point of the solution set condition is to avoid collapsing categories down to posets!