$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 8.1.2.3. Let $\operatorname{\mathcal{C}}$ be a Kan complex. Then the projection maps
\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \xleftarrow { \lambda _{-} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \xrightarrow { \lambda _{+} } \operatorname{\mathcal{C}} \]
are trivial Kan fibrations of simplicial sets.
Proof.
It follows from Corollary 8.1.1.13 that $\lambda _{-}$ and $\lambda _{+}$ are Kan fibrations. By virtue of Proposition 3.3.7.6, it will suffice to show that the fibers of $\lambda _{-}$ and $\lambda _{+}$ are contractible Kan complexes, which is an immediate consequence of Proposition 8.1.2.1 (see Corollary 4.6.7.11).
$\square$