In Mac Lane, there is a definition of an arrow between adjunctions called a map of adjunctions. In detail, if a functor $F:X\to A$ is left adjoint to $G:A\to X$ and similarly $F':X'\to A'$ is left adjoint to $G':A'\to X'$, then a map from the first adjunction to the second is a pair of functors $K:A\to A'$ and $L:X\to X'$ such that $KF=F'L$, $LG=G'K$, and $L\eta=\eta'L$, where $\eta$ and $\eta'$ are the units of the first and second adjunction. (The last condition makes sense because of the first two conditions; also, there are equivalent conditions in terms of the co-units, or in terms of the natural bijections of hom-sets).
As far as I can see, after the definition, maps of adjunctions do not appear anywhere in Mac Lane. Googling, I found this definition also in the unapologetic mathematician, again with the motivation of being an arrow between adjunctions.
But what is the motivation for defining arrows between adjunctions in the first place? I find it hard to believe that the only motivation to define such arrows is, well, to define such arrows...
So my question is: What is the motivation for defining a map of adjunctions? Where are such maps used?
Besides the unapologetic mathematician, the only places on the web where I found the term ''map of adjunctions'' were sporadic papers, from which I was not able to get an answer to my question (perhaps ''map of adjunctions'' is non-standard terminology and I should have searched with a different name?).
I came to think about this when reading Emerton's first answer
to a question about completions of metric spaces.
In that question, $X$ is metric spaces with isometric embeddings, $A$
is complete metric spaces with isometric embeddings, $X'$ is metric
spaces with uniformly continuous maps, $A'$ is complete metric
spaces with uniformly continuous maps, and $G$ and $G'$ are the
inclusions. Now, if I understand the implications of Emerton's answer
correctly, then it
is possible to choose left adjoints $F$ and $F'$ to $G$ and $G'$ such that the (non-full) inclusions $A\to A'$ and $X\to X'$ form a map of adjunctions. This made me think whether the fact that we have a map of adjunctions has any added value. Then I realized that I do not even know what was the motivation for those maps in the first place.
[EDIT: Corrected a typo pointed out by Theo Johnson-Freyd (thanks!)]