Left Inverse Mapping is Surjection

Theorem
Any left inverse of an injection 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, and by Surjection if Composite is a Surjection we see that $$g$$ is also a surjection.