Adjunctions have special characterizations whenever the intervening
categories are bicartesian (i.e. lattices in the case of posets).
These deserve to be well-known, but such situations have not been
investigated in category theory (or order theoretic residuation theory).
Some of these results can be seen in the properties of
boolean conjugates, introduced by Jónsson & Tarski (1951) as an
underlying principle in the axioms of (abstract) relation algebras.
In the restrictive setting of boolean algebras, conjugacy and
residuation are equivalent concepts (Jónsson & Tsinakis, 1993).
However, the properties of adjunctions for bicartesian categories mean that
conjugacy should instead be understood as equivalent to a generalized notion
of the prenuclei from the theory of frames and quantales.
An application of this new understanding is the correct axiomatization
for the algebra of relations within constructive set theory.
I will discuss the particulars of these connections,
and give some details of these results for the simpler case of lattices,
to avoid the prerequisite of 2-dimensional category theory.