I want to show that for every idempotent $e : B\mapsto B$ where $e \circ e = e$ (definition of idempotent/identity), we have at least a set $A$ and a section-retraction pair $s : A \mapsto B$ and $r: B \mapsto A$ such that $s \circ r = e$.
I would like prove by contradiction, by taking that there does not exist such a set or such a section-retraction pair. But this gives me too limited information for me to work with. Is there a better idea?
Hint: try using small examples for $B$ to find an idea of how it is true.
For example, try using a two element set $B:=\{a,b\}$ and letting $e\colon \{a,b\}\to\{a,b\}$ be defined by $e(x) = a$, for any $x\in\{a,b\}$. A hint for this is to try letting $A$ be some subset of $B$.