Identity Mapping is Automorphism/Semigroups

Theorem
Let $\left({S, \circ}\right)$ be a semigroup whose identity is $e$.

Then $I_S: \left({S, \circ}\right) \to \left({S, \circ}\right)$ is a semigroup automorphism.

Proof
The main result Identity Mapping is an Automorphism holds directly.

As $I_S$ is a bijection, the only element that maps to $e$ is $e$ itself.

Thus the kernel is $\left\{{e}\right\}$.