Remark 1.3.2.10. The category $\operatorname{Mon}$ of monoids (Definition 1.3.2.1) can be regarded as a subcategory of the category $\operatorname{Mon}^{\operatorname{nu}}$ of nonunital monoids (Variant 1.3.2.8). Beware that this subcategory is not full. If $M$ and $M'$ are monoids containing unit elements $e$ and $e'$, respectively, then a nonunital monoid homomorphism $f: M \rightarrow M'$ need not satisfy the identity $f(e) = e'$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$