Identity Mapping is Automorphism/Ordered Semigroups

Theorem
Let $\struct {S, \circ, \preccurlyeq}$ be an ordered semigroup.

Then $I_S: \struct {S, \circ, \preccurlyeq} \to \struct {S, \circ, \preccurlyeq}$ is a ordered semigroup automorphism.

Proof
From Identity Mapping is Semigroup Automorphism:
 * $I_S: \struct {S, \circ} \to \struct {S, \circ}$ is a semigroup automorphism.

From Identity Mapping is Order Isomorphism:
 * $I_S: \struct {S, \preccurlyeq} \to \struct {S, \preccurlyeq}$ is an order isomorphism.