Remark 5.3.8.6. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix { \operatorname{\mathcal{E}}' \ar [rr]^{F} \ar [dr]^{U'} & & \operatorname{\mathcal{E}}\ar [dl]_{U} \\ & \operatorname{\mathcal{C}}, & } \]
where $U$ and $U'$ are minimal inner fibrations. Then $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$ if and only if it is an isomorphism of simplicial sets (see Proposition 4.7.6.13).