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?

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    $\begingroup$ For me (an algebraic geometer), the natural setting would be a cartesian diagram of topoi, and in that case we would call the natural transformation the "base change" map. There are important cases where this map is an isomorphism, e.g. when the map $H$ is "proper" (this is the $q=0$ part of the proper base change theorem in SGA4). So I would try to characterize properness in category-theoretic terms. $\endgroup$ – Piotr Achinger Dec 3 '14 at 5:55
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    $\begingroup$ Btw, for those sorts of transforms of 2-cells along the adjunctions, the (probably Australian) term is "mate". $\endgroup$ – Dimitri Chikhladze Dec 3 '14 at 23:33
  • $\begingroup$ @PiotrAchinger, thanks --- I haven't thought about the problem this way. Dimitri, good point, thanks. $\endgroup$ – Michal R. Przybylek Dec 4 '14 at 18:29

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