Left Identity in Semigroup may not be Unique

Theorem
Let $\struct {S, \circ}$ be a semigroup.

Let $e_L$ be a left identity of $\struct {S, \circ}$.

Then it is not necessarily the case that $e_L$ is unique.

Proof

 * Proof by Counterexample

Let $\struct {S, \gets}$ be an algebraic structure in which the operation $\to$ is the right operation.

From Structure under Right Operation is Semigroup, $\struct {S, \to}$ is a semigroup.

From Element under Right Operation is Left Identity, every element of $\struct {S, \to}$ is a left identity.

The result follows.