Identity Mapping is Group Endomorphism

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

Then $I_G: \struct {G, \circ} \to \struct {G, \circ}$ 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.