Surjection iff Right Inverse/Non-Uniqueness

Theorem
Let $S$ and $T$ be sets such that $S \ne \varnothing$.

Let $f: S \to T$ be a surjection.

Then a right inverse of $f$ is in general not unique.

Uniqueness occurs iff $f$ is a bijection.

Proof
If $f$ is not an injection then:
 * $\exists y \in T: \exists x_1, x_2 \in S: f \left({x_1}\right) = y = f \left({x_2}\right)$

Hence we have more than one choice in $f^{-1}\left({\left\{{y}\right\}}\right)$ for how to map $g \left({y}\right)$.

This does not happen iff $f$ is an injection.

Hence the result.