The following question must have been asked dozens of times, but I do not recall any non-trivial results.

Consider an adjoint square where the arrows indicate directions of $F, G, H, K$.

$\require{AMScd} \begin{CD} \mathbb{A} @>{ G_! \dashv G}>> \mathbb{B}\\ @VVK \dashv K^*V @VVH\dashv H^*V \\ \mathbb{C} @>>{F_! \dashv F}> \mathbb{D} \end{CD}$

I have four, dual to each other, questions. Let me pick one, but be aware that I am equally interested in getting an answer to any. Assume: $H \circ G$ is naturally isomorphic to $F \circ K$. When do $K$ and $H$ form a map of adjunctions --- i.e. when is the canonical transformation between $F_! \circ H$ and $K \circ G_!$ an isomorphism?

Given a natural isomorphism $\theta \colon H \circ G \rightarrow F \circ K$ the canonical transformation $F_! \circ H \rightarrow K \circ G_!$ is defined as the following composition:

$$F_! \circ H \overset{F_! \circ H \circ \eta^G} \rightarrow F_! \circ H \circ G \circ G_! \overset{F_! \circ \theta \circ G_!} \rightarrow F_! \circ F \circ K \circ G_! \overset{\epsilon^F \circ K \circ G_!}\rightarrow K \circ G_!$$

Are there any interesting conditions that make the square a map of adjunctions?

MOTIVATION: One encounters such situations as a kind of Beck-Chevalley condition --- when one tries to prove that there exists a "global" structure in terms of "local" (pointwise) structures. In most cases, verifying the Beck-Chevalley condition is a tedious routine. However, in some rare cases, the Beck-Chevalley condition simply does not hold --- and then this fact, almost always, puzzles me greatly. Because they puzzle me, it means that I do not fully understand them. So my question really is: are there any Royal Roads to Beck-Chevalley type conditions?