Remark 5.3.8.4. Suppose we are given a pullback diagram of simplicial sets
\[ \xymatrix { \operatorname{\mathcal{E}}' \ar [d]^{U'} \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r]^{F} & \operatorname{\mathcal{C}}. } \]
If $U$ is a minimal inner fibration, then $U'$ is a minimal inner fibration. The converse holds if $F$ is surjective.