Identity Mapping is Group Endomorphism

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

Then $I_G: \left({G, \circ}\right) \to \left({G, \circ}\right)$ is a group endomorphism.

Proof
The Identity Mapping is Group Automorphism.

By definition, a group endomorphism is a group homomorphism from $G$ to itself.

A group automorphism is a group isomorphism from $G$ to itself.

As a group isomorphism is a group homomorphism which is also a bijection, the result follows by definition.