Corollary 9.2.9.18 (Exponentiation of Factorization Systems). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category equipped with a factorization system $(S_ L, S_ R)$ and let $K$ be a simplicial set. Then the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ admits a factorization system $(S_ L^{K}, S_ R^{K})$, where $S_{L}^{K}$ denotes the collection of all morphisms $f$ in $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ such that $f(v) \in S_ L$ for each vertex $v$ of $K$, and $S_{R}^{K}$ is defined similarly.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Since $S_{L}$ and $S_{R}$ contain identity morphisms and are closed under isomorphism, the collections $S_{L}^{K}$ and $S_{R}^{K}$ have the same properties. By virtue of Theorem 9.2.9.17, it will suffice to show that the restriction map
\[ D_ K: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{Fun}(K, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{Fun}(K, \operatorname{\mathcal{C}}) ) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma ) \]
is an equivalence of $\infty $-categories. This follows from Remark 4.5.1.16, since $D_ K$ is obtained by applying the functor $\operatorname{Fun}(K, \bullet )$ to the restriction map $D: \operatorname{Fun}_{LR}(\Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. $\square$