[tominated]

Not quite - think of the functor as a container for the recursive field, not the values attached to leaves. The leaf is your base recursive case and doesn't have a value to map over. Having your functor instance map over this recursive field is what allows you to write a non-recursive function that expects the field to have the already processed value from it's children!

And yeah, the compiler can derive the functor instance! I don't know the mechanism for how it works but I think it might assume the last type parameter is the 'container' field (think the values in a list or maybe)

It's not in the standard lib, but you can find a few implementations (it's usually called cata for catamorphism) available on hackage. I have used recursion-schemes and it works pretty well.

I learned about this stuff via this excellent blog series: https://blog.sumtypeofway.com/posts/introduction-to-recursio...