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

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