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

Corollary 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

\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 Using Corollary 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 (and Remark $\square$