Definition 4.8.9.2. Let $f: A \rightarrow B$ be a morphism of simplicial sets and let $n$ be an integer. We say that $f$ is categorically $n$-connective if, for every essentially $(n-1)$-categorical functor of $\infty $-categories $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \ar [r]^-{\circ f} \ar [d]^{U \circ } & \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \ar [d]^{ U \circ } \\ \operatorname{Fun}( B, \operatorname{\mathcal{D}}) \ar [r]^-{\circ f} & \operatorname{Fun}( A, \operatorname{\mathcal{D}}) } \]
is a categorical pullback square.