Kerodon

Proof. Using Corollary 5.6.7.3, we can choose a pullback diagram

where $U'$ is a cocartesian fibration of $\infty $-categories. By virtue of Remark 8.6.1.4, it will suffice to show that $U'$ admits a cartesian conjugate, which follows from Proposition 8.6.2.3. $\square$

Proof. Unwinding the definitions, we have a pullback diagram of simplicial sets

where the lower horizontal map classifies the morphism $\lambda _{+}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{+}$ and $\overline{V}'$ is given by composition with $U$. The functor $\lambda _{-}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}$ is a cocartesian fibration (Proposition 8.2.1.7) and therefore exponentiable (Proposition 5.3.6.1). It follows from Proposition 4.5.9.18 guarantees that $\overline{V}'$ is an isofibration, so that $\overline{V}$ is also an isofibration.

Let us say that a morphism $\widetilde{e}$ in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}^{+} }( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ is special if it satisfies the condition described in $(2)$. Let us identify $\widetilde{e}$ with a pair $(e, f_{e} )$, where $e: D' \rightarrow D$ is a morphism in the $\infty $-category $\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}$ and $f_{e}: \Delta ^1 \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a functor of $\infty $-categories. Let $\pi : \Delta ^1 \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^1$ be given by the projection map onto the first factor. Then $\pi $ is a cocartesian fibration, and a morphism $u$ of $\Delta ^1 \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}} \operatorname{\mathcal{D}}$ is $\pi $-cocartesian if and only if its image in $\operatorname{\mathcal{D}}_{+}$ is an isomorphism (see Proposition 8.2.1.7). If this condition is satisfied, the assumption that $\widetilde{e}$ is special guarantees that $f_{e}( u )$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}$, and is therefore $U$-cartesian (Proposition 5.1.1.9). Applying Lemma 5.3.6.11, we deduce that $\widetilde{e}$ is $\overline{V}'$-cartesian when regarded as a morphism of $\operatorname{Fun}( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$, and therefore also $\overline{V}$-cartesian when regarded as a morphism of $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{+} }( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$.

To show that $\overline{V}$ is a cartesian fibration, it will suffice to show that if $(D, f_{D} )$ is an object of $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}^{+} }( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$, then every morphism $e: C \rightarrow D$ in the $\infty $-category $\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}$ can be lifted to a special morphism $\widetilde{e}: (C, f_{C}) \rightarrow (D, f_{D} )$ in $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}^{+} }( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$. We first claim that the lifting problem

admits a solution $f_{e}$ which is $U$-right Kan extended from $\{ D\} \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}$. Using the criterion of Corollary 7.3.5.9, we are reduced to showing that if $u: X' \rightarrow X$ is a $\pi $-cocartesian morphism of $\Delta ^1 \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}$ lying over the nondegenerate edge of $\Delta ^1$, then its image in $\operatorname{\mathcal{D}}_{+}$ can be lifted to a $U$-cartesian morphism $E \rightarrow f_{D}(X)$ in $\operatorname{\mathcal{E}}$. Since the image of $u$ in $\operatorname{\mathcal{D}}_{+}$ is an isomorphism (Proposition 8.2.1.7), this follows from our assumption that $U$ is an isofibration. Note that the functor $f_{e}$ carries every $\pi $-cocartesian morphism $u$ of $\Delta ^1 \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}$ to an isomorphism in $\operatorname{\mathcal{D}}$ (if the image of $u$ in $\Delta ^1$ is degenerate, then $u$ is an isomorphism and this condition is automatically satisfied), so that $\widetilde{e} = (e, f_{e} )$ is a special morphism of $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}^{+} }( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ having target $(D, f_ D)$. This completes the proof of $(1)$.

To complete the proof of $(2)$, it will suffice to show that every $\overline{V}$-cartesian morphism $\widetilde{e} = (C, f_{C}) \rightarrow (D, f_{D})$ of $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}^{+} }( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ is special. Let $e: C \rightarrow D$ denote the image of $\widetilde{e}$ in the $\infty $-category $\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}$. Using the preceding argument, we can lift $e$ to a special morphism $\widetilde{e}': (C, f'_{C}) \rightarrow (D, f_{D} )$ of $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}^{+} }( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ Write $\widetilde{e} = (e, f_ e )$ and $\widetilde{e}' = (e, f'_{e} )$. Since $\widetilde{e}'$ is also $\overline{V}$-cartesian, Remark 5.1.3.8 guarantees that the functors $f_{e}$ and $f'_{e}$ are isomorphic. In particular, if $u$ is a morphism of $\Delta ^1 \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}$ such that $f'_{e}(u)$ is an isomorphism in $\operatorname{\mathcal{E}}$, then $f_{e}(u)$ is also an isomorphism in $\operatorname{\mathcal{E}}$. It follows that the morphism $\widetilde{e}$ is also special, as desired. $\square$

Proof. We will prove assertion $(1)$; assertions $(2)$ and $(3)$ then follow as formal consequences (see Proposition 5.1.4.17). Let us identify $\widetilde{e}$ with a pair $(e, f_{e} )$, where $e: C \rightarrow D$ is a morphism in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ and $f_{e}: \Delta ^1 \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ is a functor. Let $u: \widetilde{C} \rightarrow \widetilde{C}'$ be a morphism in the fiber $\{ C \} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$; we wish to show that $f_{C}(u)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$. Since the projection map $\Delta ^1 \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \Delta ^1$ is a cocartesian fibration, we can choose a diagram

in the $\infty $-category $\Delta ^1 \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$, where $v$ is a morphism of $\{ D \} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ and the horizontal maps are $\pi $-cocartesian. Applying the functor $f_{e}$, we obtain a diagram

in the $\infty $-category $\operatorname{\mathcal{E}}$, where the horizontal maps are isomorphisms (by virtue of our assumption that $\widetilde{e}$ is $\overline{V}$-cartesian; see Lemma 8.6.2.6). It will therefore suffice to show that $f_{D}(v)$ is $U$-cocartesian (Corollary 5.1.2.5), which follows from our assumption that $(D,f_{D})$ is an object of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$. $\square$

Proof. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ denote the twisted arrow fibration of Example 8.2.0.2. Recall that $(a)$ is equivalent to the following pair of conditions:

$(a_1)$

For every object $C \in \operatorname{\mathcal{C}}$, the restriction of $T$ to the fiber over the vertex $\{ \operatorname{id}_{C} \} \subseteq \operatorname{Tw}(\operatorname{\mathcal{C}})$ determines an equivalence of $\infty $-categories $T_{C}: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} \rightarrow \{ C \} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.

$(a_2)$

Let $(e',e)$ be an edge of the fiber product $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. If $e'$ is a $U^{\dagger }$-cartesian morphism of $\operatorname{\mathcal{E}}^{\dagger }$, then $T(e',e)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$.

Unwinding the definitions, we see that $F$ factors through $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ if and only if $T$ satisfies the following weaker version of $(a_2)$:

$(b_0)$

Let $(e',e)$ be an edge of the fiber product $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. If $e'$ is a degenerate edge of $\operatorname{\mathcal{E}}^{\dagger }$, then $T(e',e)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$.

If this condition is satisfied, then we have a commutative diagram

where the vertical maps are cartesian fibrations (Lemma 8.6.2.7). Using Theorem 5.1.6.1, we see that $F$ is an equivalence if and only if it satisfies the following further conditions:

$(b_1)$

For each object $C \in \operatorname{\mathcal{C}}$, the functor $F$ restricts to an equivalence of $\infty $-categories

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

$(b_2)$

The functor $F$ carries $U^{\dagger }$-cartesian morphisms of $\operatorname{\mathcal{E}}^{\dagger }$ to $V$-cartesian morphisms of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$.

Let $C$ be an object of $\operatorname{\mathcal{C}}$. Unwinding the definitions, we can identify the fiber

with the $\infty $-category $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})$. Since $\operatorname{id}_{C}$ is initial when viewed as an object of the $\infty $-category $\{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ (Proposition 8.1.2.1), Proposition 5.3.1.21 guarantees that the evaluation map

is a trivial Kan fibration. Moreover, the composition of this evaluation map with the functor $F_{C}$ coincides with the functor $T_{C}$ appearing in condition $(a_1)$. It follows that conditions $(a_1)$ and $(b_1)$ are equivalent.

Using the characterization of $V$-cartesian morphisms supplied by Lemmas 8.6.2.6 and 8.6.2.7, we can reformulate $(b_2)$ more concretely as follows:

$(b'_2)$

Let $(e',e)$ be an edge of the fiber product $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. If $e'$ is a $U^{\dagger }$-cartesian morphism of $\operatorname{\mathcal{E}}^{\dagger }$ and $\lambda _{+}(e)$ is an isomorphism in $\operatorname{\mathcal{C}}$, then $T(e',e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}$.

To complete the proof, it will suffice to show that the functor $T$ satisfies $(a_2)$ if and only if it satisfies both $(b_0)$ and $(b'_2)$. The implication $(a_2) \Rightarrow (b_0)$ is immediate, and the implication $(a_2) \Rightarrow (b'_2)$ follows from Corollary 5.1.1.9. The reverse implication follows from Proposition 8.6.1.13. $\square$

Proof of Proposition 8.6.2.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. It follows from Lemma 8.6.2.7 that the projection map $V: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cartesian fibration. We wish to show that the evaluation map

exhibits $V$ as a cartesian conjugate of $U$. This follows from Proposition 8.6.2.8, since the identity automorphism of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ is an equivalence of $\infty $-categories. $\square$

Proof of Proposition 8.6.2.10. Let $\lambda = ( \lambda _{-}, \lambda _{+} ): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ denote the twisted arrow coupling of Example 8.2.0.2, and let $V: \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ denote the composition of $\lambda _{+}$ with projection onto the second factor. Note that $V$ factors as a composition

where the first map is a left fibration (since it is a pullback of $\lambda $, which is a left fibration by virtue of Proposition 8.1.1.15), and the last map is a cocartesian fibration (since it is a pullback of the projection map $\operatorname{\mathcal{E}}^{\dagger } \rightarrow \Delta ^0$). It follows that $V$ is a cocartesian fibration, and that a morphism of $\operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ is $V$-cocartesian if and only if its image in $\operatorname{\mathcal{E}}^{\dagger }$ is an isomorphism. In particular, our hypotheses on $T$ guarantees that it carries $V$-cocartesian morphisms of $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ to $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$.

Fix an object $C \in \operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{E}}^{\dagger }(C)$ denote the fiber $V^{-1} \{ C\} = \operatorname{\mathcal{E}}' \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ C\} $, so that projection onto the middle factor gives a map $U^{\dagger }_{C}: \operatorname{\mathcal{E}}^{\dagger }(C) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ C\} $. Note that $U^{\dagger }_{C}$ is a pullback of $U^{\dagger }$. It follows that $U^{\dagger }_{C}$ is a cartesian fibration, and that a morphism of $\operatorname{\mathcal{E}}^{\dagger }(C)$ is $U^{\dagger }_{C}$-cartesian if and only if its image in $\operatorname{\mathcal{E}}^{\dagger }$ is $U^{\dagger }$-cartesian (Remark 5.1.4.6). Let $W_ C$ denote the collection of morphisms of $\operatorname{\mathcal{E}}^{\dagger }(C)$ which satisfy this condition, so that $W = \bigcup _{C \in \operatorname{\mathcal{C}}} W_ C$. Note that $T$ restricts to a functor $T^{C}: \operatorname{\mathcal{E}}^{\dagger }(C) \rightarrow \operatorname{\mathcal{E}}_{C}$. By virtue of Proposition 6.3.5.2, it will suffice to verify the following (for each object $C \in \operatorname{\mathcal{C}}$):

$(\ast _ C)$

The functor $T^{C}$ exhibits the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ as a localization of $\operatorname{\mathcal{E}}^{\dagger }(C)$ with respect to $W_{C}$.

Let $\operatorname{\mathcal{K}}$ denote the full subcategory of $\operatorname{Tw}( \operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} $ whose objects are isomorphisms $D \rightarrow C$. By virtue of Proposition 8.1.2.1, $\operatorname{\mathcal{K}}$ can also be described as the full subcategory of $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} $ spanned by its initial objects. It follows that $\operatorname{\mathcal{K}}$ is a coreflective subcategory of $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} $ (Example 6.2.2.5). Let $\operatorname{\mathcal{E}}^{\dagger }_0(C) \subseteq \operatorname{\mathcal{E}}^{\dagger }(C)$ denote the inverse image of $\operatorname{\mathcal{K}}$ under $U^{\dagger }_{C}$, so that $\operatorname{\mathcal{E}}^{\dagger }_0(C)$ is a coreflective subcategory of $\operatorname{\mathcal{E}}^{\dagger }(C)$ (Proposition 6.2.2.24). Using Lemma 6.2.2.16, we can choose a functor $L: \operatorname{\mathcal{E}}^{\dagger }(C) \rightarrow \operatorname{\mathcal{E}}^{\dagger }_0(C)$ and a natural transformation $\epsilon : L \rightarrow \operatorname{id}_{ \operatorname{\mathcal{E}}^{\dagger }(C)}$ which exhibits $L$ as a $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$-coreflection functor. Our assumption on $T$ guarantees that the functor $T^{C}$ carries each element of $W_{C}$ to an isomorphism in $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$, so that $\epsilon $ induces an isomorphism of functors $(T^{C}|_{ \operatorname{\mathcal{E}}^{\dagger }_{0}(C)} \circ L) \rightarrow T^ C$. Since the Kan complex $\operatorname{\mathcal{K}}$ is contractible (Corollary 4.6.7.14), the inclusion map $\{ \operatorname{id}_{C} \} \hookrightarrow \operatorname{\mathcal{K}}$ is a homotopy equivalence of Kan complexes, and therefore induces an equivalence of $\infty $-categories $\operatorname{\mathcal{E}}^{\dagger }(C) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} \simeq \operatorname{\mathcal{E}}^{\dagger }_{0}(C)$ (Corollary 4.5.2.29). Since the composition

is an equivalence of $\infty $-categories, we conclude that the functor $T^{C}|_{ \operatorname{\mathcal{E}}^{\dagger }_{0}(C) }$ is also an equivalence of $\infty $-categories. To complete the proof of $(2_ C)$, it will suffice to show that the functor $L$ exhibits $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$ as a localization of $\operatorname{\mathcal{E}}^{\dagger }(C)$ with respect to $W_ C$ (see Remark 6.3.1.19). Let $W^+_{C}$ denote the collection of morphisms $v$ of $\operatorname{\mathcal{E}}^{\dagger }(C)$ such that $L(v)$ is an isomorphism in $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$. By virtue of the preceding arguments, this is equivalent to the requirement that $T(v)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$; in particular, assumption $(1)$ guarantees that $W_ C$ is contained in $W^{+}_{C}$. Conversely, if $u: Y \rightarrow Z$ is a morphism of $\operatorname{\mathcal{E}}^{\dagger }(C)$ which belongs to $W^{+}_ C$, then we can choose a commutative diagram

where $u$ and $w$ exhibit $X$ as $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$-coreflections of the objects $Y$ and $Z$, respectively, and therefore belong to $W_{C}$. We are therefore reduced to showing that the functor $L$ exhibits $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$ as a localization of $\operatorname{\mathcal{E}}^{\dagger }(C)$ with respect to $W^{+}_{C}$, which is a special case of Example 6.3.3.7. $\square$

Proof. We will prove assertions $(1)$, $(2)$, and $(3)$; assertion $(4)$ is then a formal consequence. Applying Proposition 8.6.2.10 to $T_0$ we deduce that the functor $Q$ is fully faithful, and that its essential image is spanned by those objects $G \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})$ which satisfy the following condition:

$(\ast )$

Let $w = (w',w'')$ be a morphism of $\operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}})$. If $w'$ is $U^{\dagger }$-cartesian and $w''$ has degenerate image in $\operatorname{\mathcal{C}}$, then $G(w)$ is an isomorphism in $\operatorname{\mathcal{E}}$.

Applying Proposition 8.6.2.8 we see that $Q^{\dagger }$ is also fully faithful, and that its essential image consists of those $G$ which satisfy the following:

$(\ast ^{\dagger })$

Let $w = (w',w'')$ be a morphism of $\operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}})$. If $w'$ is a degenerate edge of $\operatorname{\mathcal{D}}^{\dagger }$, then $G(w)$ is a $V$-cocartesian morphism of $\operatorname{\mathcal{E}}$.

This proves assertion $(1)$.

To prove $(2)$, we must show that an object $F \in \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ carries $U$-cocartesian morphisms of $\operatorname{\mathcal{D}}$ to $V$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ if and only if the induced map

satisfies condition $(\ast ^{\dagger })$. The “only if” direction follows from our assumption that $T_0$ exhibits $U^{\dagger }$ as a cartesian conjugate of $U$. For the converse, assume that $Q(F) = F \circ T_0$ satisfies condition $(\ast ^{\dagger })$ and let $f: X \rightarrow Y$ be a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$, having image $e: C \rightarrow D$ in $\operatorname{\mathcal{C}}$. Choose an object $X' \in \operatorname{\mathcal{D}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C \} $ such that $T_0(X')$ is isomorphic to $X$ as an object of the $\infty $-category $\{ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$. Let $e_{L}: \operatorname{id}_{C} \rightarrow e$ be the morphism in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ introduced in Example 8.1.3.6, and regard $w = (\operatorname{id}_{X'}, e_{L} )$ as an edge of $\operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. Our assumption on $T_0$ guarantees that $T_0(w)$ is a $U$-cocartesian edge of $\operatorname{\mathcal{D}}$ lying over $e$, and is therefore isomorphic to $f$ (as an object of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}})$). It will therefore suffice to show that $(F \circ T_0)(w)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$, which follows from condition $(\ast ^{\dagger })$.

To prove $(3)$, we must show that an object $F^{\dagger } \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{D}}^{\dagger }, \operatorname{\mathcal{E}}^{\dagger } )$ carries $U^{\dagger }$-cartesian morphisms of $\operatorname{\mathcal{D}}^{\dagger }$ to $V^{\dagger }$-cartesian morphisms of $\operatorname{\mathcal{E}}^{\dagger }$ if and only if the induced map

satisfies condition $(\ast ^{\dagger })$. The “only if” direction follows immediately from our assumption that $T$ exhibits $V^{\dagger }$ as a cartesian conjugate of $V$, and the reverse implication follows from the criterion of Remark 8.6.1.14. $\square$