Example 8.6.1.3. Let $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}^{\dagger }$ be $\infty $-categories. Set $\operatorname{\mathcal{C}}= \Delta ^0$ and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}$ denote the projection maps. Then a functor
\[ T: \operatorname{\mathcal{E}}^{\dagger } \simeq \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}} \]
exhibits $U^{\dagger }$ are a cartesian conjugate of $U$ if and only if it an equivalence of $\infty $-categories. In particular, $U^{\dagger }$ is a cartesian conjugate of $U$ if and only if the $\infty $-category $\operatorname{\mathcal{E}}^{\dagger }$ is equivalent to $\operatorname{\mathcal{E}}$.