Permutation of Set is Automorphism of Set under Left Operation

Theorem
Let $S$ be a set.

Let $\struct {S, \gets}$ be the algebraic structure formed from $S$ under the left operation.

Let $f$ be a permutation on $S$.

Then $f$ is an automorphism of $f$.

Proof
We have that $f$ is a permutation and so a fortiori a bijection.

It remains to show that $f$ is a homomorphism.

So, let $a, b \in S$ be arbitrary.

We have:

The result follows.

Also see

 * Permutation of Set is Automorphism of Set under Right Operation