Remark 9.4.6.3 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{F} \ar [dr]_{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{U'} \\ & \operatorname{\mathcal{C}}, & } \]
where the vertical maps are inner fibrations and $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$ (see Definition 5.1.7.1). Then $U$ is flat if and only if $U'$ is flat.