Definition 8.6.1.1 (Conjugate Fibrations). Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. Let $\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the projection maps of Notation 8.1.1.6. We say that a morphism of simplicial sets $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ exhibits $U^{\dagger }$ as a cartesian conjugate of $U$ if the following conditions are satisfied:
- $(0)$
The diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \lambda _{+} } & \operatorname{\mathcal{C}}} \]is commutative.
- $(1)$
For every vertex $C \in \operatorname{\mathcal{C}}$, restricting $T$ to the inverse image of the vertex $\operatorname{id}_{C} \in \operatorname{Tw}(\operatorname{\mathcal{C}})$ determines an equivalence of $\infty $-categories $T_{C}: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$.
- $(2)$
Let $e$ be an edge of the simplicial set $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}})$. If the image of $e$ in $\operatorname{\mathcal{E}}^{\dagger }$ is $U^{\dagger }$-cartesian, then $T(e)$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
We say that $U^{\dagger }$ is a cartesian conjugate of $U$ if there exists a morphism $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ which exhibits $U^{\dagger }$ as a cartesian conjugate of $U$.