Remark 1.3.2.11. Let $M$ be a nonunital monoid, and let $M^{+} = M \cup \{ e\} $ be the enlargement of $M$ obtained by formally adjoining a new element $e$. Then the multiplication on $M$ extends uniquely to a monoid structure on $M^{+}$ having unit element $e$. Moreover, if $M'$ is any other monoid, then the restriction map $f \mapsto f|_{M}$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Monoid homomorphisms $f: M^{+} \rightarrow M'$} \} \ar [d] \\ \{ \textnormal{Nonunital monoid homomorphisms $f_0: M \rightarrow M'$} \} . } \]
Consequently, the inclusion functor $\operatorname{Mon}\hookrightarrow \operatorname{Mon}^{\operatorname{nu}}$ has a left adjoint, given on objects by the construction $M \mapsto M^{+}$.