1) John, can you recommend a book that covers monads as a next book after Fong & Spivak (especially if it includes applications such as functional programming)?

*****

2) I have a question that arose from trying to transfer Chapter 1's discussion of adjunctions to functors. For a preorder adjunction \\(f \dashv g\\), we saw that by setting \\(b = f(a)\\), we could derive \\(a \leq g(f(a))\\).

Doing the same thing for categories results in a natural isomorphism between

\[ \textrm{Hom}\_\mathcal{A}(a,R(L(a))) \qquad\textrm{and}\qquad \textrm{Hom}\_\mathcal{B}(L(a),L(a)) \]

Since the second hom-set always contains at least an identity morphism, the first hom-set must contain at least one element. That non-emptiness is the analog of the inequality for preorders.

But here we have an isomorphism between \\(\textrm{id}_{L(a)}\\) and some morphism \\(s:a \rightarrow R(L(a))\\). Since identity morphisms are very special, it feels like there should be something special about its isomorphic partner. Is that true?

I wasn't able to find something special. I tried following the naturality conditions, but no unusual properties jumped out at me.

*****

2) I have a question that arose from trying to transfer Chapter 1's discussion of adjunctions to functors. For a preorder adjunction \\(f \dashv g\\), we saw that by setting \\(b = f(a)\\), we could derive \\(a \leq g(f(a))\\).

Doing the same thing for categories results in a natural isomorphism between

\[ \textrm{Hom}\_\mathcal{A}(a,R(L(a))) \qquad\textrm{and}\qquad \textrm{Hom}\_\mathcal{B}(L(a),L(a)) \]

Since the second hom-set always contains at least an identity morphism, the first hom-set must contain at least one element. That non-emptiness is the analog of the inequality for preorders.

But here we have an isomorphism between \\(\textrm{id}_{L(a)}\\) and some morphism \\(s:a \rightarrow R(L(a))\\). Since identity morphisms are very special, it feels like there should be something special about its isomorphic partner. Is that true?

I wasn't able to find something special. I tried following the naturality conditions, but no unusual properties jumped out at me.