Proposition 5.1.7.15. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [rr]^-{F} \ar [dr]_{U} & & \operatorname{\mathcal{D}}\ar [dl]^{V} \\ & \operatorname{\mathcal{E}}, & } \]
where $U$ and $V$ are cartesian fibrations. Then $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$ if and only if the following conditions are satisfied:
- $(1)$
For every vertex $E \in \operatorname{\mathcal{E}}$, the induced map $F_{E}: \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}\rightarrow \{ E \} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories.
- $(2)$
The morphism $F$ carries $U$-cartesian edges of $\operatorname{\mathcal{C}}$ to $V$-cartesian edges of $\operatorname{\mathcal{D}}$.