Expressing that a functor is natural

Question

The Haskell List: Type -> Type constructor implements the Functor typeclass with function fmap f = map f. This functor that applies the morphism f to each element of a list works, but there are other equally valid functors, for example fmap f = map f . rev which first reverses the list before mapping over it. Is there a concept in category theory that expresses the fact that the first choice of functor is in some sense "more natural" for this type constructor?

Answer 1

Being a functor requires two properties:

1. fmap id = id
2. fmap (f . g) = fmap f . fmap g

The definition fmap fails the first condition:

fmap id = map id . reverse = reverse != id

So that's not a functor at all!

And it even fails the second condition:

fmap (f . g) = map (f . g) . reverse
             = map f . map g . reverse
             = map f . fmap g
             = reverse . map f . reverse . fmap g
             = reverse . fmap f . fmap g
             != fmap f . fmap g

Which is quite clear if you think about it:

• fmap (f . g) reverses the list once and then maps its elements via f . g
• fmap f . fmap g reverse the list, maps via g, then reverses the list again and then maps via f!