This seems interesting, but I think there are some assumptions that have been left out? What prevents us from taking G and g to be constants, which would give solutions for any f?
I am indeed leaving out some assumptions, mainly because I am not yet convinced of which assumptions are “right”. The simplest assumption—used by Aczel—is that G and g are monotonic. But that’s usually chosen more for mathematical convenience than for any principled reason, as far as I can tell. We certainly want some assumptions which rule out the trivial solution, but I’m not sure what they should be.