$ \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\PS}{\mathcal P} \newcommand{\Idl}{\operatorname{Idl}} $
There are many situations where one encounters a square of things which are related by pairs of adjoint functors:
where $f\dashv g$ means that $f$ is left adjoint to $g$.
Example: $f\colon R\to S$ is a morphism of rings, $X=\PS R$ and $Y=\PS S$ are the powersets of $R$ and $S$, and $A=\Idl(R)$ and $B=\Idl(B)$ are the lattices of ideals of $R$ and $S$; the adjunctions are "ideal generated by a powerset $\dashv$ ideal viewed as subset", "extension of ideal $\dashv$ contraction of ideal, and two instances of "direct image of subset $\dashv$ inverse image of subset".
Alternatively, one of the adjunctions may be a Galois correspondence instead; for example, a ring map $f\colon R\to S$ induces adjunctions between $\mathcal P\Spec(R)$ and $\mathcal P\Spec(S)$, as well as between the lattices of ideals of $R$ and $S$; but $\mathcal P(\Spec(S))$ and $\Idl(S)$ are related by a Galois correspondence ($I$ and $V$) rather than an adjunction.
What is the succinct categorical way to describe such squares (of either type)? I imagine the answer is something to do with double categories... On a more practical level: I have several examples of such "compatible squares of adjoints/correspondences", such as those listed above, and am in search of a formal definition of "compatible square".
- Given a quadruplet of adjunctions, what conditions should I demand for them to be "compatible"?
- What equalities (or just natural transformations) can I infer from such a square?
For the most part I only care about the case of posets, but of course a general categorical answer would be most satisfying.