Corollary 2.1.5.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital lax monoidal functor. Then $F$ admits a unit $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F(\mathbf{1}_{\operatorname{\mathcal{C}}} )$ if and only if it has both a left unit and a right unit. In this case, the unit $\epsilon $ is unique.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$