By the way, if we want to prove a monotone function between _posets_ has no left (resp. right) adjoint, one option is to use the [Adjoint Functor Theorem for posets](https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1) and show that it doesn't preserve all joins (resp. meets).

Conversely, if our monotone map between posets _does_ preserve all joins (resp. meets) then it _does_ have a left (resp. right) adjoint. But in this case it's often just as easy to guess the desired adjoint and prove it obeys the definition of adjoint.