What is the motivation for maps of adjunctions?

Here is an example of how one might have stumbled upon the definition of a map of adjunctions. Suppose that you are working on a research project with a collaborator. Let's call her Jane for the sake of argument. On the first day you and Jane realize that your joint research project depends partly on knowing whether a certain functor F has a right adjoint. It also depends on taking that supposed right adjoint and putting it to good use. So you really need to know what that right adjoint is. You and Jane call it a day, and agree to continue working the next day.

That night both of you are independently inspired. You wake up in the middle of the night an jot down some notes. The next morning you and Jane meet to discuss what you've each figured out. Fantastic news! Both of you have found the right adjoint to F. You immediately begin planing how you are going to solve XYZ-Big-Problem with this fabulous right adjoint. After the celebratory mood wares away, you and Jane realize with some horror the truth. Your right adjoint G is not the same as Jane's right adjoint G'. They are different functors and the adjunction structure maps are different.

Whatever are you to do? Which one should you use?

Fortunately Jane has a flash of insight. We know that two functors can be isomorphic, what about adjunctions? After thinking about this some more, you and Jane figure out that a morphism of adjunctions should be a morphism of functors which preserves the adjunction structure. You try to do this is the simplest way possible and BAM! You've rediscovered the notion of morphism of adjunction. You notice that while G and G' are not the same functor (and hence not the same adjunction) they are isomorphic adjunctions. Whew!

But now you and Jane start to seriously worry. You have your functor F and you know that your right adjoint G and Jane's right adjoint G' are isomorphic. But what happens when Prof. X comes along with his right adjoint G''? Will it be isomorphic to G and G'? Given F, how unique is its right adjoint? Even if G, G', and G'' are all isomorphic adjunctions there might be some monodromy, i.e. the isomorphism, $$G \to G' \to G'' \to G$$ might theoretically fail to be the identity.

Then you read a little farther in MacLane and you find this theorem (I'm rephrasing it with some terminology which is in vogue.

Theorem: Given a functor F, the category of right adjoints to F (with their adjunction data and with morphisms of adjunctions as morphisms) is either empty or is a contractible category (i.e. it is equivalent to a terminal category i.e. any two objects are isomorphic and that isomorphism is unique).

So you can stop worrying. Any other right adjoint G'' that Prof. X brings to you will in fact be (uniquely) isomorphic to the one you discovered.

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 $X$.

The 2-category of categories, adjunctions, and conjugate natural transformations (i.e., maps of adjunctions between the same categories) is used in an approach to modal type theory in Adjoint Logic with a 2-Category of Modes.

The general 2-categorical account of the double category of categories, adjunctions and maps of adjunctions is given by mates.