Example 2.4.2.12 (The Conjugate of a Simplicial Category). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We define a new simplicial category $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ as follows:
The objects of $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ are the objects of $\operatorname{\mathcal{C}}_{\bullet }$.
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }) = \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, we have an equality of simplicial sets
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{c}} }( X, Y)_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }^{\operatorname{op}}; \]here the right hand side denotes the opposite of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ (Construction 1.4.2.2).
For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }) = \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the composition law
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{c}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{c}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{c}}}(X,Z)_{\bullet } \]on $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ is obtained from the composition law on $\operatorname{\mathcal{C}}_{\bullet }$ by passing to opposite simplicial sets.
We will refer to $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ as the conjugate of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$.