Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 2.4.2.13. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ denote the conjugate simplicial category (Example 2.4.2.12). Then, when regarded as a simplicial object of $\operatorname{Cat}$, the conjugate simplicial category $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ is given by the functor

\[ \operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { \mathrm{Op}} \operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { [n] \mapsto \operatorname{\mathcal{C}}_{n} } \operatorname{Cat}; \]

here $\mathrm{Op}$ denotes the involution of $\operatorname{{\bf \Delta }}$ described in Notation 1.3.2.1. In particular, the underlying ordinary categories of $\operatorname{\mathcal{C}}_{\bullet }$ and $\operatorname{\mathcal{C}}_{\bullet }^{\operatorname{c}}$ are the same.