Bijection iff Left and Right Cancellable

Theorem

Let $f$ be a mapping.

Then $f$ is a bijection if and only if $f$ is both left cancellable and right cancellable.

Proof

Follows directly from:

Injection iff Left Cancellable: $f$ is an injection if and only if $f$ is left cancellable
Surjection iff Right Cancellable: $f$ is a surjection if and only if $f$ is right cancellable.

$\blacksquare$