How to abstract over a “back and forth” transformation?

Question

Consider this example (from https://codereview.stackexchange.com/questions/23456/crtitique-my-haskell-function-capitalize):

import Data.Char

capWord [] = []
cap         


        

Answer 1

Your lifted is actually the same as dimap from Data.Profunctor:

onWords = dimap words unwords
capitalize = onWords (map capWord)

That might not be the direction of generalization you thought about. But look at the type of the equivalent function in Control.Functor from category-extras:

dimap :: Bifunctor f (Dual k) k k => k b a -> k c d -> k (f a c) (f b d)

This version generalizes it over everything which is both a QFunctor and a co-PFunctor. Not that useful in everyday scenarios, but interesting.