Let $\displaystyle \left({S, \circ}\right)$ be a semigroup.

Let $\displaystyle u \in S$ be such that:

$\displaystyle \forall a \in S: \exists x, y \in S: u \circ x = a = y \circ u$

Prove that $\displaystyle \left({S, \circ}\right)$ has an identity element.