Remark 2.1.7.13. Let $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{A}}'$ be a lax monoidal functor between monoidal categories. Then the construction of Remark 2.1.7.4 determines a functor $\operatorname{Cat}(\operatorname{\mathcal{A}}) \rightarrow \operatorname{Cat}(\operatorname{\mathcal{A}}')$. In the special case where $\operatorname{\mathcal{A}}' = \operatorname{Set}$ and $F$ is the functor $A \mapsto \underline{\operatorname{Hom}}_{\operatorname{\mathcal{A}}}( \mathbf{1}, \operatorname{\mathcal{A}})$ corepresented by the unit object $\mathbf{1} \in \operatorname{\mathcal{A}}$, we obtain a forgetful functor
which assigns to each (small) $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}$ its underlying ordinary category (Example 2.1.7.5).