Injection iff Left Cancellable

Theorem
A mapping $f$ is an injection iff $f$ is left cancellable.

Proof
From the definition: a mapping $f: Y \to Z$ is left cancellable iff:


 * $\forall X: \forall g_1: X \to Y, g_2: X \to Y: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$

Also see

 * Surjection iff Right Cancellable