Proposition 2.1.2.3. Let $M$ be a nonunital monoid, let $e$ be an element of $M$, and let $\ell _{e}: M \rightarrow M$ denote the function given by the formula $\ell _{e}(x) = ex$. The following conditions are equivalent:

- $(a)$
The element $e$ is a left unit of $M$: that is, $\ell _{e}$ is the identity function from $M$ to itself.

- $(b)$
The element $e$ is idempotent (that is, it satisfies $ee = e$) and the function $\ell _{e}: M \rightarrow M$ is a bijection.

- $(c)$
The element $e$ is idempotent and the function $\ell _{e}: M \rightarrow M$ is a monomorphism.