\begin{definition}[Definition:Free Monoid]
Let $S$ be a set.
A '''free monoid over $S$''' is a monoid $M$ together with a mapping $i: S \to M$, subject to:
:For all monoids $N$, for all mappings $f: S \to N$, there is a unique monoid homomorphism $\bar f: M \to N$, such that:
::$\bar f \circ i = f$
This condition is called the '''universal (mapping) property''' or '''UMP''' of the '''free monoid over $S$'''.
\end{definition}