Exercise 8.3.6.6. Verify that Construction 8.3.6.5 is well-defined. That is, for every linearly ordered set $J$ and every morphism $\sigma : \operatorname{N}_{\bullet }(J) \rightarrow \operatorname{Tw}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) )$, show that the simplicial functor $F_{\sigma }$ admits a unique extension $\widetilde{F}_{\sigma }: \operatorname{Path}[J^{\triangleleft }]_{\bullet } \rightarrow \operatorname{Kan}$ which satisfies conditions $(a)$ and $(b)$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$