Corollary 9.2.8.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing $2$-simplices $\sigma $ and $\tau $. Suppose that the initial edge $f = d^{2}_{2}(\sigma )$ is left orthogonal to the final edge $g = d^{2}_{0}(\tau )$. Then the restriction map $\theta : \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }(\sigma , \tau ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) }(f,g)$ is a trivial Kan fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Proposition 3.3.7.6, it will suffice to show that every fiber of $\theta $ is a contractible Kan complex. Let $D: \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ denote the functor given by precomposition with the inclusion map $\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \hookrightarrow \Delta ^2$, and let $E: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be given by evaluation at the final vertex $1 \in \Delta ^1$. Let $\gamma : \Delta ^2 \times \Delta ^1 \rightarrow \Delta ^2$ be the morphism of simplicial sets given on vertices by the formula $\gamma (i,j) = \begin{cases} 1 & \end{cases}$ $\square$