Corollary 8.1.2.20. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, which we regard as an object of the twisted arrow $\infty $-category $\operatorname{Tw}(\operatorname{\mathcal{C}})$. Then:
The morphism $f$ is $U$-cocartesian if and only if it is $V$-initial, where $V$ denotes the induced map
\[ \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \{ U(X) \} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}}). \]The morphism $f$ is $U$-cartesian if and only if it is $V'$-initial, where $V'$ denotes the induced map
\[ \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{D}}) \times _{\operatorname{\mathcal{D}}} \{ U(Y) \} . \]