Kerodon

Remark 2.2.4.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a lax functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. Then, for each object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, we can regard $F_{X,X}: \underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X) \rightarrow \underline{\operatorname{End}}_{\operatorname{\mathcal{D}}}( F(X) )$ as a lax monoidal functor from $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$ (endowed with the monoidal structure of Remark 2.2.1.7) to $\underline{\operatorname{End}}_{\operatorname{\mathcal{D}}}( F(X) )$: the tensor and unit constraints on $F_{X,X}$ are given by the composition and identity constraints on $F$, respectively. If $F$ is a functor, then $F_{X,X}$ is a monoidal functor.