Definition 2.1.5.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. A lax monoidal structure on $F$ is a nonunital lax monoidal structure $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ (Definition 2.1.4.3) for which there exists a unit $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$.
A lax monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a pair $(F, \mu )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is functor and $\mu $ is a lax monoidal structure on $F$. In this case, we will refer to the morphism $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$ as the unit of $F$.