Identity of Inverse Completion of Commutative Monoid

Theorem
Let $\left({S, \circ}\right)$ be a commutative monoid whose identity is $e$.

Let $\left ({C, \circ}\right) \subseteq \left({S, \circ}\right)$ be the subsemigroup of cancellable elements of $\left({S, \circ}\right)$.

Let $\left({T, \circ'}\right)$ be an inverse completion of $\left({S, \circ}\right)$.

Then $e \in T$ is the identity for $\circ'$.

Proof
Let $e$ be the identity for $\circ$.

Let $e = x \circ' y^{-1}$, where $x \in S, y \in C$.

Then:

Thus $e = y^{-1} \circ' y$, and $y^{-1} \circ' y$ is the identity for $\circ'$.