Remark 8.6.4.4 (Base Change). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets and $\lambda : \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ be a left fibration. The following conditions are equivalent:
- $(a)$
The left fibration $\lambda $ exhibits $U^{\vee }$ as a cocartesian dual of $U$ (in the sense of Definition 8.6.4.1).
- $(b)$
For every morphism of simplicial sets $\operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$, form a diagram of pullback squares
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0^{\vee } \ar [d] \ar [r]^-{U^{\vee }_{0}} & \operatorname{\mathcal{C}}' \ar [d] & \operatorname{\mathcal{E}}_0 \ar [l]_{ U_0 } \ar [d] \\ \operatorname{\mathcal{E}}\ar [r]^-{ U^{\vee } } & \operatorname{\mathcal{C}}& \operatorname{\mathcal{E}}. \ar [l]_{U} } \]Then the induced map
\[ \lambda _0: (\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{\mathcal{E}}^{\vee }_{0} \times _{ \operatorname{\mathcal{C}}_0 } \operatorname{\mathcal{E}}_{0} \]exhibits $U^{\vee }_0$ as a cocartesian dual of $U_{0}$.
Moreover, it suffices to verify condition $(b)$ in the special case where $\operatorname{\mathcal{C}}_0 = \Delta ^1$ is the standard $1$-simplex.