Left Inverse Mapping is Surjection

Theorem

Let $f: S \to T$ be an injection.

Let $g: T \to S$ be a left inverse of $f$.

Then $g$ is a surjection.

Proof

Let $f: S \to T$ be an injection.

Then from Injection iff Left Inverse there exists at least one left inverse $g: T \to S$ of $f$ such that $g \circ f = I_S$.

$I_S$ is a surjection.

Thus $g \circ f$ is a surjection.

By Surjection if Composite is Surjection it follows that $g$ is also a surjection.

$\blacksquare$