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.

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.