Theorem 9.6.7.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$. If $S_{L}$ is left orthogonal to $S_{R}$, then the restriction functor
\[ D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma ) \]
is fully faithful. The converse holds if $S_{L}$ and $S_{R}$ contain all identity morphisms of $\operatorname{\mathcal{C}}$.
Proof of Theorem 9.6.7.2.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$. Suppose first that $S_{L}$ is left orthogonal to $S_{R}$. In this case, Proposition 9.6.7.4 guarantees that the restriction map
\[ \theta : \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }(\sigma , \sigma ') \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) }( d^{2}_{1}(\sigma ), d^{2}_{1}(\sigma ') ) \]
is a homotopy equivalence whenever $\sigma $ belongs to $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$ and $\sigma '$ belongs to $\operatorname{Fun}_{R}( \Delta ^2, \operatorname{\mathcal{C}})$. It follows that the functor
\[ D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma ) \]
is fully faithful.
We now prove the converse. Assume that $D$ is fully faithful and that $S_{L}$ and $S_{R}$ contain all identity morphisms of $\operatorname{\mathcal{C}}$. Take $f \in S_ L$ and $g \in S_ R$, and let $\widetilde{f} = s^{1}_{1}(f)$ and $\widetilde{g} = s^{1}_{0}(g)$ denote the degenerate $2$-simplices of $\operatorname{\mathcal{C}}$ appearing in the proof of Corollary 9.6.8.9. Since $S_{L}$ and $S_{R}$ contain all identity morphisms, we can view $\widetilde{f}$ and $\widetilde{g}$ as objects of the $\infty $-category $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$. Our assumption that $D$ is fully faithful guarantees that the Kan fibration $\operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }( \widetilde{f}, \widetilde{g} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }( f,g )$ is a homotopy equivalence, so that $f$ is left orthogonal to $g$ by virtue of Remark 9.6.7.5.
$\square$