Kerodon

Example 2.2.4.12 (Enriched Categories as Lax Functors). Let $S$ be a set, and let $\operatorname{\mathcal{E}}_{S}$ denote the indiscrete category with object set $S$: that is, the objects of $\operatorname{\mathcal{E}}_{S}$ are the elements of $S$, and $\operatorname{Hom}_{\operatorname{\mathcal{E}}_{S}}(X,Y)$ is a singleton for every pair of elements $X,Y \in S$. Regard $\operatorname{\mathcal{E}}_{S}$ as a (strict) $2$-category having only identity $2$-morphisms (Example 2.2.0.6). Let $\operatorname{\mathcal{C}}$ be a monoidal category, and let $B\operatorname{\mathcal{C}}$ be its delooping (Example 2.2.2.5). Unwinding the definitions, we see that lax functors $F: \operatorname{\mathcal{E}}_{S} \rightarrow B \operatorname{\mathcal{C}}$ (in the sense of Definition 2.2.4.5) can be identified with $\operatorname{\mathcal{C}}$-enriched categories having object set $S$ (in the sense of Definition 2.1.7.1).