Remark 4.8.6.16. Let $n$ be an integer and suppose we are given a categorical pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [d]^{F'} \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r]^-{G} & \operatorname{\mathcal{D}}. } \]
If $F$ is essentially $n$-categorical, then $F'$ is essentially $n$-categorical. The converse holds if $G$ is full and essentially surjective. See Corollary 4.8.5.29.