I'm in chapter 4 of Bird's very interesting Thinking Functionally with Haskell and he has a problem at the end of the chapter where he lists these equations

map f . take n = take n . map f

map f . reverse = reverse . map f

map f . sort = sort . map f

map f . filter p = map fst . filter snd . map (fork (f,p))

filter (p . g) = map (invertg) . filter p . map g

reverse . concat = concat . reverse . map reverse

filter p . concat = concat . map (filter p)

adding this caveat for the 3rd equation

iff x <= y <=> f x <= f y

and this for the 4th equation

fork (f,g) x = (f x, g x)

and for the 5th invertg satisfies invertg . g = id

My confusion is over the commutative-ness of most of this but only anecdotally. With the particularly dense

map f . filter p = map fst . filter snd . map (fork (f,p))

We have

> :t (map myF . filter myP)

Integral b => [b] -> [b]

> :t (map fst . filter snd . map (myFork (myF,myP)))

Integral b => [b] -> [b]

Is there anything universal to be drawn from these anecdotal examples of seeming commutativity? My breakdown of the third equation shows the same type definition for both sides. Is this a way to find equality? All in all, Bird doesn't indicate that there are any underlying truths, just "almost" commutativity.