In order to see if I really understand this theorem and the proofs, I decided to apply it to a “simple” example with monotone maps f and g between two power sets: f: P{a,b,c}} -> P{1,2,3,4} and g: P{1,2,3,4} -> P{a,b,c}.

But because the two power sets are of different sizes, I don’t see how to construct monotone maps f and g between them.

I’m embarrassed to ask such a basic question, but can anyone give me some suggestions?