Corollary 8.6.2.9 (05JG)

Corollary 8.6.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $V^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. Suppose we are given a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [rr]^{ T } \ar [dr] & & \operatorname{\mathcal{E}}\ar [dl]^{ V } \\ & \operatorname{\mathcal{C}}& } \]

which exhibits $V^{\dagger }$ as a cartesian conjugate of $V$. Then, for every morphism of simplicial sets $U^{\dagger }: \operatorname{\mathcal{D}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$, composition with $T$ induces a fully faithful functor

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{D}}^{\dagger }, \operatorname{\mathcal{E}}^{\dagger } ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}}). \]

The essential image is spanned by those diagrams $Q: \operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ having the property that, for every vertex $D \in \operatorname{\mathcal{D}}^{\dagger }$, $Q$ carries every edge of $\{ D\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ to a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$.