I think it's clearer to think of map as a function (a -> b) -> (f a -> f b) i.e. one which lifts function application over values of the functor. This is

I think it's clearer to think of map as a function (a -> b) -> (f a -> f b) i.e. one which lifts function application over values of the functor. This is then analagous to the role of `f` for mapping types, and corresponds to the concept of a functor in the category of types. This relationship is harder to see in the tupled version. There's a similar argument for e.g. =< :: (a -> m b) -> (m a -> m b)