Kerodon

Example 2.1.7.5 (The Underlying Category of an Enriched Category). Let $\operatorname{\mathcal{A}}$ be a monoidal category and let $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{Set}$ be the functor given by $F( A ) = \operatorname{Hom}_{\operatorname{\mathcal{A}}}( \mathbf{1}, A)$, endowed with the lax monoidal structure of Example 2.1.5.16. If $\operatorname{\mathcal{C}}$ is a category enriched over $\operatorname{\mathcal{A}}$, then we can apply the construction of Remark 2.1.7.4 to obtain a $\operatorname{Set}$-enriched category, which we can identify with an ordinary category (Example 2.1.7.2). We will refer to this category as the underlying category of the $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}$, and we will generally abuse notation by denoting it also by $\operatorname{\mathcal{C}}$. Concretely, this underlying category has the same objects as the enriched category $\operatorname{\mathcal{C}}$, with morphism sets given by the formula $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) = \operatorname{Hom}_{\operatorname{\mathcal{A}}}( \mathbf{1}, \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )$.