I am interested in the category $A$ of adjunctions that induce a monad $c : C \to C$ where $C$ is a poset. (The description of $A$ is in a previous math.se post.) For a general $C$, of course, $A$ could be very complicated, but I would imagine that for a poset $C$ it is more feasible to give an explicit description of $A$.
Is there a good source from which I can learn about such a description of this category? Even if this is not possible for general posets, is it feasible for special posets (lattices, locales, etc)?