$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 4.8.8.24. Let $n$ be an integer, and suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [rr]^{F} \ar [dr] & & \operatorname{\mathcal{D}}\ar [dl] \\ & \operatorname{\mathcal{E}}& } \]
where the vertical maps are inner fibrations. If $F$ is categorically $(n+1)$-connective, then it induces an equivalence of $\infty $-categories $F': \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{E}})} \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{D}}/\operatorname{\mathcal{E}})}$.
Proof.
We have a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d] & \operatorname{\mathcal{D}}\ar [d] \\ \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{E}})} \ar [r]^-{F'} & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{D}}/\operatorname{\mathcal{E}})}. } \]
Here $F$ is categorically $(n+1)$-connective by assumption, and the vertical maps are categorically $(n+1)$-connective by virtue of Corollary 4.8.8.19. Applying Proposition 4.8.7.12, we see that the functor $F'$ is also categorically $(n+1)$-connective. We also have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{E}})} \ar [rr]^{F'} \ar [dr] & & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{D}}/\operatorname{\mathcal{E}})} \ar [dl] \\ & \operatorname{\mathcal{E}}, & } \]
where the vertical maps are essentially $n$-categorical (Proposition 4.8.8.14). Using Remark 4.8.6.8, we see that $F'$ is also essentially $n$-categorical. Using Remark 4.8.5.11, we see that $F'$ is an equivalence of $\infty $-categories.
$\square$