A set map is onto exactly when it splits on the right.Suggest a slogan
Let \(f\colon X\to Y\) be a map of sets. Then \(f\) is surjective, that is, for every \(y\in Y\) there exists \(x\in X\) with \(f(x)=y\), if and only if \(f\) has a right inverse: a map \(g\colon Y\to X\) with \(f(g(y))=y\) for every \(y\in Y\).
If \(g\) is a right inverse, then each \(y\in Y\) equals \(f(g(y))\), so \(f\) is surjective. Conversely, if \(f\) is surjective, then each fiber \(f^{-1}(y)\) is nonempty, and the axiom of choice gives a map \(g\colon Y\to X\) with \(g(y)\in f^{-1}(y)\) for every \(y\); then \(f(g(y))=y\).
What condition on inverses characterises a surjective function?