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 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.