A function that takes itself as an input and returns different function with the same property?

Question

Consider a set of functions $F$ = $\{f$ such that $f: F \rightarrow F\}$. For example identity function $f_{id} \in F$ since $f_{id}(f_{id}) = f_{id} \in F$. Are there some more interesting functions $f \in F$ such that $f(f)=f'$, $f \neq f'$, $f' \in F$ so that $f'(f')=f'' \in F,\; f''(f'')=f''' \in F ,\;...$? Where can I learn about such functions?

inspiration: The Cognitive-Theoretic Model of the Universe

Answer 1

I don't think you can properly define such a set $F$ unless $F$ consists only of $f_{id}$. To see this, let $f,g\in F$ with $f\ne g$. For each subset $G$ of $F$ we can define a function mapping $G$ to $f$ and $F\setminus G$ to $g$. In particular $F$ has a subset of the same cardinality as the power set of $F$. However the power set of $F$ has strictly larger cardinality than $F$.