Remark 4.8.1.9. Suppose we are given a categorical pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d] & \operatorname{\mathcal{D}}\ar [d] \\ \overline{\operatorname{\mathcal{C}}} \ar [r]^-{\overline{F}} & \overline{\operatorname{\mathcal{D}}}. } \]
If $\overline{F}$ is essentially surjective, then $F$ is essentially surjective. This follows from Proposition 4.5.3.15 and Corollary 3.5.1.24.