Kerodon

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Proposition 5.1.6.14. 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 $\operatorname{\mathcal{D}}$.

Proof. By virtue of Proposition 5.1.6.9, we may assume without loss of generality that $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex, so that $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$ if and only if it is an equivalence of $\infty $-categories (Corollary 5.1.6.8). Since $U$ and $V$ are isofibrations (Example 4.4.1.6), the desired result follows from Theorem 5.1.5.1. $\square$