Surjection iff Right Inverse

Theorem
A mapping $f: S \to T, S \ne \varnothing$ is a surjection :
 * $\exists g: T \to S: f \circ g = I_T$

where:
 * $g$ is a mapping
 * $I_T$ is the identity mapping on $T$.

That is, if $f$ has a right inverse.

In general, that right inverse is not unique.

Uniqueness occurs $f$ is a bijection.

Also see

 * Injection iff Left Inverse