Right Inverse Mapping is Injection

Theorem
Any right inverse of a surjection is an injection.

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

Then from Surjection iff Right Inverse there exists at least one right inverse $h: T \to S$ of $f$ such that $f \circ h = I_T$.

$I_T$ is an injection.

Thus $f \circ h$ is an injection, and by Injection if Composite is an Injection we see that $h$ is also a injection.

Also see

 * Left Inverse Mapping is Surjection