Remark 4.7.9.3. Let $\kappa $ be an uncountable cardinal and suppose we are given a pullback diagram of simplicial sets
\[ \xymatrix { \operatorname{\mathcal{E}}' \ar [d]^{U'} \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}, } \]
where $U$ and $U'$ are inner fibrations. If $U$ is essentially $\kappa $-small, then $U'$ is essentially $\kappa $-small. If $U$ is locally $\kappa $-small, then $U'$ is locally $\kappa $-small.