Remark 8.6.7.3 (04FB): Comparison with the Conjugate

Remark 8.6.7.3 (Comparison with the Conjugate). Let $\operatorname{\mathcal{E}}$ be a simplicial category. Recall that the conjugate $\operatorname{\mathcal{E}}^{\operatorname{c}}$ is a simplicial category having the same objects, with morphism spaces given by $\operatorname{Hom}_{\operatorname{\mathcal{E}}^{\operatorname{c}}}(X,Y)_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet }^{\operatorname{op}}$ (see Example 2.4.2.12). Then there is a canonical isomorphism of simplicial categories $\operatorname{\mathcal{E}}^{\asymp } \xrightarrow {\sim } ( \operatorname{\mathcal{E}}^{\operatorname{c}} )^{\asymp }$, which is the identity on objects and given on morphism spaces by the isomorphisms

\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}^{\asymp }}( X,Y )_{\bullet } = \operatorname{Tw}( \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet } ) \simeq \operatorname{Tw}( \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet }^{\operatorname{op}} ) = \operatorname{Hom}_{ (\operatorname{\mathcal{E}}^{\operatorname{c}} )^{\asymp } }( X,Y)_{\bullet } \]

described in Remark 8.1.1.7. Composing this isomorphism with the forgetful functor $(\operatorname{\mathcal{E}}^{\operatorname{c}})^{\asymp } \rightarrow \operatorname{\mathcal{E}}^{\operatorname{c}}$, we obtain a forgetful functor $\pi ^{\operatorname{c}}: \operatorname{\mathcal{E}}^{\asymp } \rightarrow \operatorname{\mathcal{E}}^{\operatorname{c}}$. If $\operatorname{\mathcal{E}}$ is locally Kan, then Proposition 8.6.7.2 guarantees that $\pi ^{\operatorname{c}}$ is a weak equivalence of simplicial categories. We therefore obtain equivalences of $\infty $-categories

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}) \xleftarrow {\sim } \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}^{\asymp } ) \xrightarrow {\sim } \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}^{c} ). \]