Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.8.7.18. Let $n$ be an integer and let $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{B}}$ and $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty $-categories. Suppose that $F$ is categorically $n$-connective and that $G$ is essentially $(n-1)$-categorical. Then the diagram of $\infty $-categories

4.86
\begin{equation} \begin{gathered}\label{equation:orthogonality-categoricity-connectivity} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}) \ar [r]^-{\circ F} \ar [d]^{G \circ } & \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{C}}) \ar [d]^{ G \circ } \\ \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \ar [r]^-{\circ F} & \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{D}}) } \end{gathered} \end{equation}

is a categorical pullback square.

Proof of Corollary 4.8.7.18. Using Corollary 4.8.7.16 we can reduce to the case where $F$ is a monomorphism which is bijective on simplices of dimension $\leq n$. The desired result now follows from Corollary 4.8.6.20 (and Remark 4.8.7.19). $\square$