Injection iff Left Inverse

Theorem
A mapping $f: S \to T, S \ne \varnothing$ is an injection :
 * $\exists g: T \to S: g \circ f = I_S$

where $g$ is a mapping.

That is, $f$ has a left inverse.

Also see

 * Surjection iff Right Inverse


 * Conditions for Uniqueness of Left Inverse Mapping: in general the left inverse is not unique.