Proposition 9.2.9.4. Let $n$ be an integer, let $S_{L}$ denote the collection of all categorically $n$-connective functors, and let $S_{R}$ denote the collection of all essentially $(n-1)$-categorical functors. Then the pair $(S_ L, S_ R)$ is a factorization system on the $\infty $-category $\operatorname{\mathcal{QC}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We first observe that $S_{R}$ is closed under the formation of relative diagonals: that is, if a functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is essentially $(n-1)$-categorical, then the relative diagonal of $G$ (formed in the $\infty $-category $\operatorname{\mathcal{QC}}$) has the same property. Using Exercise 7.6.3.13, we can identify the relative diagonal of $G$ with the inclusion map $\iota : \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$. For $n \geq 1$, it follows from Variant 4.8.6.15 that $\iota $ is essentially $(n-2)$-categorical, and therefore also essentially $(n-1)$-categorical (Remark 4.8.6.6). If $n \leq 0$, then the functor $G$ is fully faithful, so $\iota $ is an equivalence of $\infty $-categories.
It follows from Remarks 4.8.5.16, 4.8.5.17, and 4.8.5.18 that $S_{L}$ and $S_{R}$ are invariant under isomorphism. Theorem 4.8.8.3 asserts that every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ admits a factorization $\operatorname{\mathcal{C}}\xrightarrow {F_{L}} \operatorname{\mathcal{D}}\xrightarrow {F_{R}} \operatorname{\mathcal{E}}$, where $F_{L}$ belongs to $S_{L}$ and $F_{R}$ belongs to $S_{R}$. We will complete the proof by showing that $S_{L}$ is left orthogonal to $S_{R}$. By virtue of (the dual of) Corollary 9.2.7.21, it will suffice to show that $S_{L}$ is weakly left orthogonal to $S_{R}$: that is, every lifting problem
9.22
\begin{equation} \begin{gathered}\label{equation:factorization-system-on-QCat} \xymatrix { \operatorname{\mathcal{A}}\ar [r] \ar [d]^{F} & \operatorname{\mathcal{C}}\ar [d]^{G} \\ \operatorname{\mathcal{B}}\ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{QC}}$ admits a solution, provided that $F$ is categorically $n$-connective and $G$ is essentially $(n-1)$-categorical. By virtue of Corollary 5.6.5.18, we may assume that (9.22) arises from a commutative diagram in the category of simplicial sets. Using Corollary 4.5.2.23, we can further assume that $F$ is a monomorphism of simplicial sets and that $G$ is an isofibration. In this case, the lifting problem (9.22) already admits a solution in the category of simplicial sets: see Corollary 4.8.7.18 and Remark 4.8.7.19. $\square$