Kerodon

Remark 2.4.2.14. Let $\operatorname{\mathcal{C}}$ be a strict $2$-category and let $\operatorname{\mathcal{C}}_{\bullet }$ denote the associated simplicial category (Example 2.4.2.8). Then the conjugate simplicial category $(\operatorname{\mathcal{C}}_{\bullet })^{\operatorname{c}}$ can be identified with the simplicial category $(\operatorname{\mathcal{C}}^{\operatorname{c}})_{\bullet }$ associated to the conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$ of Construction 2.2.3.4. In particular, if $\operatorname{\mathcal{C}}$ is an ordinary category, then we have a canonical isomorphism $\operatorname{\mathcal{C}}_{\bullet }^{\operatorname{c}} \simeq \operatorname{\mathcal{C}}_{\bullet }$.