Thank you for this. I find explanations that eschew category theory bizarre, this really cleared things up for me (having spent a lot of time with functors mathematically, and almost no time with them in a programming context).
Why do you get a composition of functions by applying a functor to individual functions? It seems like a functor is just something you can map over. When a functor is mapped over the end result is another functor. Not a function nor a composition of functions.
fmap (g . f) = (fmap g) . (fmap f)
and
fmap id = id
where (.) :: (b -> c) -> (a -> b) -> (a -> c) is the function composition operator and id :: a -> a the identity function. Those are the laws the last two sentences try to phrase in english. In mathematics a functor F between categories C and D maps objects in C to objects in D and any morphism f: X -> Y in C to a morphism F f : F X -> F Y in D, in such a way that for morphisms f : X -> Y and g : Y -> Z in C one has F (g . f) = F g . F f and F id_X = id_{F X}. So you see Haskell and math notation are almost identical, although you can express the laws only as compiler rules in Haskell.
A function is a Functor. ;)
If you have a function f and a function g, fmap f g = h, and h is a function. A more common way to write this is function composition of f and g, which can be written: f . g. The dot (.) is function composition. Which means that fmap f g = f . g, which means that fmap = (.).
> When a functor is mapped over the end result is another functor. Not a function nor a composition of functions.
But they was describing the case when the Functor in question is a function.