Remark 4.6.2.19. Suppose we are given a categorical pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [d]^{F'} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}. } \]
If $F$ is essentially surjective, then $F'$ is essentially surjective. This follows from Proposition 4.5.2.14 and Corollary 3.5.1.24.