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Corollary 8.6.6.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories, let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration of $\infty $-categories, and suppose we are given a commutative diagram

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{ U } \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}. } \]

The following conditions are equivalent:

$(1)$

The functor $T$ exhibits $U^{\dagger }$ as a cartesian conjugate of $U$ (in the sense of Definition 8.6.1.1).

$(2)$

The functor $T$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ with respect to $W$, where $W$ is the collection of all morphisms $w = (w', w'')$ where $w'$ is a $U'$-cartesian morphism of $\operatorname{\mathcal{E}}^{\dagger }$ and $w''$ is a morphism of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ whose image in $\operatorname{\mathcal{C}}$ is degenerate.

Proof. We will show that $(2)$ implies $(1)$; the reverse implication follows from Proposition 8.6.2.12. Using Corollary 8.6.6.3, we can choose a cocartesian fibration $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ and a commutative diagram

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T'} \ar [d] & \operatorname{\mathcal{E}}' \ar [d]^{ U'} \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}} \]

which exhibits $U^{\dagger }$ as a cartesian conjugate of $U'$. Assume that condition $(2)$ is satisfied, so that we have a commutative diagram

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ) \ar [r]^-{T \circ } \ar [d]^{U \circ } & \operatorname{Fun}( (\operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}))[W^{-1}], \operatorname{\mathcal{E}}' ) \ar [d]^{ U' \circ } \\ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}) \ar [r]^-{T \circ } & \operatorname{Fun}( (\operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}))[W^{-1}], \operatorname{\mathcal{C}}), } \]

where the horizontal maps are equivalences of $\infty $-categories and the vertical maps are isofibrations (Corollary 4.4.5.6). Applying Corollary 4.5.2.32, we deduce that the map

\[ (\circ T): \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}}' ) \]

is fully faithful, and that its essential image consists of those functors $\operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}'$ which carry each morphism of $W$ to an isomorphism in $\operatorname{\mathcal{E}}'$. We may therefore assume without loss of generality that $T' = F \circ T$ for some functor $F \in \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}')$. Proposition 8.6.2.12 implies that $T'$ exhibits $\operatorname{\mathcal{E}}'$ as a localization of $\operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ with respect to $W$. It follows that $F$ is an equivalence of $\infty $-categories (Remark 6.3.1.19), so that $T$ also exhibits $U^{\dagger }$ as a cartesian conjugate of $U$. $\square$