# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Example 2.1.2.8. Let $M$ be a nonunital monoid, regarded as a (strict) nonunital monoidal category having only identity morphisms (Example 2.1.1.3). Then the converse of Example 2.1.2.7 holds: a pair $(\mathbf{1}, \upsilon )$ is a unit structure on $M$ (in the sense of Definition 2.1.2.5) if and only if $\mathbf{1}$ is a unit element of $M$ and $\upsilon = \operatorname{id}_{ \mathbf{1} }$. This is a restatement of Corollary 2.1.2.4.