[Ken Scambler wrote:](https://forum.azimuthproject.org/discussion/comment/16256/#Comment_16256):

> I have a question: are there interesting things I can learn about (co)monads by studying adjoint monotone functions composed this way or that way?

Yes! I'm assuming from your question that you know preorders are secretly categories of a special sort and Galois connection are secretly adjoint functors of a special sort. So, you are in a position to think about things like this:

**Puzzle:** if we think of a preorder \\(A\\) as a category, what does a monad on \\(A\\) amount to?

Usually a monad on \\(A\\) consists of a functor \\(T : A \to A\\) together with some natural transformations obeying some properties. So, when \\(A\\) is a preorder, we know a monad on \\(A\\) will consist of a monotone function \\(T: A \to A\\) equipped with some extra bells and whistles. But preorders are so simple compared to categories in general that a lot of these bells and whistles will become trivial! But not everything: a monad on a preorder is more than just an arbitrary monotone function \\(T : A \to A\\). It's actually something incredibly interesting and important. And you can have a lot of fun figuring this out yourself.

Once you figure this out, there's more fun to be had by seeing what comonads are like in this context,

and by studying monads and comonads that come from composing adjoint monotone functions this way or that way.

> I have a question: are there interesting things I can learn about (co)monads by studying adjoint monotone functions composed this way or that way?

Yes! I'm assuming from your question that you know preorders are secretly categories of a special sort and Galois connection are secretly adjoint functors of a special sort. So, you are in a position to think about things like this:

**Puzzle:** if we think of a preorder \\(A\\) as a category, what does a monad on \\(A\\) amount to?

Usually a monad on \\(A\\) consists of a functor \\(T : A \to A\\) together with some natural transformations obeying some properties. So, when \\(A\\) is a preorder, we know a monad on \\(A\\) will consist of a monotone function \\(T: A \to A\\) equipped with some extra bells and whistles. But preorders are so simple compared to categories in general that a lot of these bells and whistles will become trivial! But not everything: a monad on a preorder is more than just an arbitrary monotone function \\(T : A \to A\\). It's actually something incredibly interesting and important. And you can have a lot of fun figuring this out yourself.

Once you figure this out, there's more fun to be had by seeing what comonads are like in this context,

and by studying monads and comonads that come from composing adjoint monotone functions this way or that way.