Why do the types in `(fmap . fmap) sum Just [1, 2, 3]` work?

Question

I'm having the time of my life reading the wonderful Haskell Programming from first principles and I came by the following example that I'm just not able to take apart (Page 1286 e-reader):

Prelude> (fmap . fmap) sum Just [1, 2, 3]
Just 6

It is obvious to me how the following works:

Prelude> fmap sum $ Just [1,2,3]
Just 6

And I already manually deconstructed (fmap . fmap) to understand how the types work. But when thinking about this as "lifting twice" it doesn't make sense, since I'm lifting over both the Just and List data constructors.

I typed out the following in ghci:

Prelude> :t (fmap . fmap)
(fmap . fmap)
  :: (Functor f, Functor f1) => (a -> b) -> f1 (f a) -> f1 (f b)

Prelude> :t (fmap . fmap) sum
(fmap . fmap) sum
  :: (Num b, Foldable t, Functor f, Functor f1) =>
     f1 (f (t b)) -> f1 (f b)

Prelude> :t (fmap . fmap) sum Just
(fmap . fmap) sum Just :: (Num b, Foldable t) => t b -> Maybe b

I don't understand how to derive the last output. When feeding (fmap . fmap) sum the Just data constructor, How does the compiler know to replace both f1 and f for Maybe? After I'll get a good answer here, how could I have figured it out myself?

Answer 1

That isn't lifting over both Maybe and List (that would be (fmap . fmap) sum (Just [1,2,3]), which has a type problem), but over the function type (->) and Maybe.

Just :: a -> Maybe a
     -- ((->) a) (Maybe a)
     -- f (g a)   for f ~ ((->) a)  and  g ~ Maybe

(fmap . fmap) :: (a   -> b) -> f (g a  ) -> f (g b)
     -- Num x => ([x] -> x) -> f (g [x]) -> f (g x)
     -- Num x => ([x] -> x) -> ([x] -> Maybe [x]) -> [x] -> Maybe x
     --          ^             ^                     ^
     --          sum           Just                  [1,2,3]