Mapping from Group Element to Inner Automorphism is Homomorphism

Theorem
Let $G$ be a group.

Let $\kappa: G \to \Aut G$ be the mapping from $G$ to the automorphism group of $G$ defined as:
 * $\forall x \in G: \map \kappa x := \kappa_x$

where $\kappa_x$ is the inner automorphism on $x$:
 * $\forall g \in G: \map {\kappa_x} g = x g x^{-1}$

Then $\kappa$ is a homomorphism.

Proof
Let $x, y \in G$.

By definition of automorphism group, we have that:
 * $\map \kappa x \map \kappa y = \kappa_x \circ \kappa_y$

where $\circ$ denotes composition of mappings.

Then $\forall g \in G$:

And so:
 * $\forall g \in G: \map {\kappa_{x y} } g = \map {\kappa_x \circ \kappa_y} g$

Thus by definition of $\kappa$:
 * $\map \kappa x \map \kappa y = \map \kappa{x y}$

demonstrating that $\kappa$ is a homomorphism.