Corollary 4.8.5.23. 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 $n$-faithful. Then the diagram of $\infty $-categories
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\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.