Injection has Surjective Left Inverse Mapping/Proof 2

Theorem
Let $f: S \to T$ be a injection.

Then there is a surjection $g: T \to S$ such that $g \circ f = I_S$.

Proof
By Injection iff Left Inverse, $f$ has a left inverse $g: T \to S$.

By Left Inverse Mapping is Surjection, $g$ is a surjection.