$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.1.9.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $(S_ L, S_ R)$ be a factorization system on $\operatorname{\mathcal{C}}$, and let $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}})$ be the full subcategory of Notation 9.1.8.1. Then the restriction map
\[ D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma ) \]
is a trivial Kan fibration.
Proof.
Condition $(1)$ of Definition 9.1.9.1 guarantees that $D$ is surjective on objects, and condition $(2)$ guarantees that $D$ is fully faithful (Theorem 9.1.8.2). Applying the criterion Theorem 4.6.2.20, we deduce that $D$ is an equivalence of $\infty $-categories. Condition $(3)$ of Definition 9.1.9.1 guarantees that the full subcategory $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ is replete, so that $D$ is an isofibration of $\infty $-categories (see Corollary 4.4.5.3). Applying Proposition 4.5.5.20, we conclude that $D$ is a trivial Kan fibration.
$\square$