Let $G$ be a groupoid. An element $e \in G$ is called a left identity element if $ex = x$ for all $x \in G$ Similarly, $e$ is a right identity element if $xe = x$ for all $x \in G$

An element which is both a left and a right identity is an identity element.

A groupoid may have more than one left identify element: in fact the operation defined by $x y = y$ for all $x, y \in G$ defines a groupoid (in fact, a semigroup) on any set $G$ and every element is a left identity.

But as soon as a groupoid has both a left and a right identity, they are necessarily unique and equal. For if $e$ is a left identity and $f$ is a right identity, then $f = ef = e$