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](http://unapologetic.wordpress.com/2007/07/30/transformations-of-adjoints/),
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](https://mathoverflow.net/questions/11622/what-is-the-right-universal-property-of-the-completion-of-a-metric-space).
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!)]



What is the motivation for maps of adjunctions?
One of the applications of adjoint functors is to compose them to get a monad (or comonad, depending on the order in which you compose them). A map of adjoint functors gives rise to a map of monads. So one might ask: what are maps of monads good for? Many algebraic categories (such as abelian groups, rings, modules) can be described as categories of algebras over a monad, others (for example in Arakelov geometry) are most easily described in such a way. A map of monads then gives functors between the categories of algebras over these objects.


Here is a concrete example from topology: Let $E$ be a connective generalized multiplicative homology theory, and let $H = H(-;\pi\_0E)$ be ordinary homology with coefficients in $\pi\_0E$. There exists a map $E \to H$ inducing an isomorphism on $\pi\_0$. For a spectrum $X$, the functor $\underline{E}\colon X \mapsto E \wedge X$ gives rise to a monad, and similarly for $H$, thus we get a morphism of monads $\underline{E} \to \underline{H}$. The completion $X\hat{{}\_E}$ of a spectrum $X$ at $E$ is defined to be the totalization of the cosimplicial spaces obtained by iteratively applying $\underline{E}$ to $X$. The monad map gives a natural map $X\hat{{}\_E} \to X\hat{{}\_H}$ which turns out to be an equivalence for connective $X$.


