Definition 8.1.10.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pushouts. We will say that an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a dual Beck-Chevalley fibration if the following conditions are satisfied:
- $(1)$
The morphism $U$ is a cartesian fibration.
- $(2)$
The morphism $U$ is a cocartesian fibration.
- $(3)$
Suppose we are given a morphism $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{E}}$, which we display informally as a diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d]^{g} & X_0 \ar [d]^{g'} \\ X_1 \ar [r]^-{f'} & X_{01}. } \]Assume that $f$ is $U$-cartesian, that $g'$ is $U$-cocartesian, and that $U(\sigma )$ is a pushout square in $\operatorname{\mathcal{C}}$. Then $f'$ is $U$-cartesian if and only if $g$ is $U$-cocartesian.