Theorem 9.2.8.2. 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 9.2.8.2.
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 Corollary 9.2.8.8 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 $\operatorname{\mathcal{C}}$; we wish to show that $S_{L}$ is left orthogonal to $S_{R}$. Suppose we are given a lifting problem $\tau :$
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur]^{s} & Y } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$, where $f$ belongs to $S_{L}$ and $g$ belongs to $S_{R}$. We wish to show that the solution space $\operatorname{Sol}(\tau )$ is contractible. Let $\widetilde{f} = s^{1}_{1}(f)$ and $\widetilde{g} = s^{1}_{0}(g)$ denote the degenerate $2$-simplices of $\operatorname{\mathcal{C}}$ defined in Notation 9.2.8.3. Since $S_{L}$ and $S_{R}$ contain all identity morphisms, we can view $\widetilde{f}$ and $\widetilde{g}$ as objects of the $\infty $-category $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$. Assumption $(1)$ 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. Lemma 9.2.8.7 supplies a homotopy equivalence of $\operatorname{Sol}( \tau )$ with the fiber $\operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }( \widetilde{f}, \widetilde{g} )_{\tau }$, which is contractible by virtue of Proposition 3.3.7.6.
$\square$