Remark 9.4.6.4. Suppose we are given a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [d]^{U'} \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r]^-{F} & \operatorname{\mathcal{C}}. } \]
If $U$ is a flat inner fibration, then $U'$ is a flat inner fibration. The converse holds if $F$ is surjective on $2$-simplices.