Yes, this is reasonable, but it might be useful to be a bit more explicit about the categories involved. An instance of Functor is a functor which maps objects and arrows in the category of Hask (of Haskell types and functions) to... objects and arrows in the category of Hask. It is an endofunctor. The object mapping must map a type to a type and therefore must have the kind * -> *. This means that it is a type constructor, e.g., Maybe. The arrow mapping must map arrows to arrows, which means it must map functions to functions. fmap provides this mapping and the Functor laws in Haskell are of course equivalent to the laws from category theory. And while the inclusion of bottom is problematic for Hask, it is still possible to reason about it in a "morally correct" way ( http://www.cse.chalmers.se/~nad/publications/danielsson-et-al-popl2006.pdf).