The principal types and definitions are:
map :: (a -> b) -> [a] -> [b]
map _ []     = []
map f (x:xs) = f x : map f xs

foldr :: (a -> b -> b) -> b -> [a] -> b
foldr _ e []     = e
foldr f e (x:xs) = f x (foldr f e xs)

(.) :: (b -> c) -> (a -> b) -> a -> c
(.) f g x = f (g x)
The map function preserves the list shape while changing element type from a to b. The right fold replaces the empty list by e and each constructor by f, whose second argument and result therefore share the accumulator type b. Function composition requires the output type of g to agree with the input type of f. No Haskell type class constraints arise: these definitions never compare elements or perform arithmetic on them.
Take h = (.):
h :: (b -> c) -> (a -> b) -> a -> c
h f g x = f (g x)
This is its standalone principal type; in the right fold, c specializes to the type of a function from the accumulator to itself.
For fixed f, g and e, prove equality pointwise by structural induction on the list. At [] both sides return e. For x:xs, assume the equality for xs and calculate:
foldr f e (map g (x:xs))
  = f (g x) (foldr f e (map g xs))
  = f (g x) (foldr (h f g) e xs)
  = h f g x (foldr (h f g) e xs)
  = foldr (h f g) e (x:xs)
The second equality is the induction hypothesis. Function extensionality gives the required equality of composed functions. This is an instance of the right-fold fusion law.
For the full lazy Haskell interpretation, include a bottom value for an undefined list spine: both sides are bottom there. The same constructor argument covers every finite approximation with an undefined tail. The Scott continuity of the list operations then extends equality to their limits, including infinite lists. Ordinary induction on finite total lists alone proves only that restricted case.
Let x :: a and g :: a -> b, as required by the inner application g x. Using g x as the next argument of that same g forces b = a. Thus g (g x) :: a. Applying that result to x requires a function type a -> c, giving
(1)
This equation fails the occurs check: the type variable a occurs inside the type with which it would be unified. These constraints have no solution in the Hindley-Milner type system. Therefore the definition has no type in ordinary Haskell. The obstruction is to type inference, independently of which particular values might later be supplied.
The expression parses as g . (g x). Initially write g :: a -> r and x :: a. In the function composition, g x must itself be a function whose result can be passed to g. Hence r = b -> a for a fresh type b, so
g :: a -> b -> a
g x :: b -> a
g . g x :: b -> b -> a
Consequently the principal type is
f :: (a -> b -> a) -> a -> b -> b -> a
One can check the constraints by expanding the composition:
f g x y z = g (g x y) z
Here g x y has type a, and the second application accepts it and the final b-typed argument z. The two type variables remain independent; no Haskell type class constraints are required.

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