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8.6.2 Existence of Conjugate Fibrations

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Our goal in this section is to show that $U$ admits a cartesian conjugate $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ (Corollary 8.6.2.4). For this purpose, we will need to construct a commutative diagram

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

We begin by considering the universal example of such a diagram.

Notation 8.6.2.1. Let $\operatorname{\mathcal{D}}$ be a simplicial set equipped with a morphism $\lambda = (\lambda _{-}, \lambda _{+} ): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$. For every simplicial set $\operatorname{\mathcal{E}}$, we let $\operatorname{Fun}( \operatorname{\mathcal{D}}/\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ denote the relative exponential of Construction 4.5.9.1. For $n \geq 0$, we will identify $n$-simplices of $\operatorname{Fun}( \operatorname{\mathcal{D}}/\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ with pairs $(\sigma , f)$, where $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}$ and $f: \Delta ^ n \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a morphism of simplicial sets. Suppose that we are also given a morphism of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}_{+}$. In this case, we let $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{+} }( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ denote the simplicial subset of $\operatorname{Fun}( \operatorname{\mathcal{D}}/\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ whose $n$-simplices are pairs $(\sigma , f)$ which satisfy the additional condition that the diagram

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^ n \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}\ar [r]^-{f} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{D}}\ar [r]^-{ \lambda _{+} } & \operatorname{\mathcal{D}}_{+} } \]

is commutative.

Construction 8.6.2.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and let $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ be the simplicial set given in Notation 8.6.2.1. Unwinding the definitions, we see that vertices of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ can be identified with pairs $(C, f_{C} )$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $f_{C}$ is a morphism which fits into a commutative diagram

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

We let $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ denote the full simplicial subset of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$, spanned by those pairs $(C, f_{C} )$ where the morphism $f_{C}$ carries each edge of $\{ C\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ to a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$. By construction, the simplicial set $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ is equipped with a projection map $V: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and an evaluation map $\operatorname{ev}: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$, given on vertices by the construction $(C, f_{C}, u: C \rightarrow C') \mapsto f_{C}(u)$.

We can now formulate the main result of this section:

Proposition 8.6.2.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Then the projection map $V: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ of Construction 8.6.2.2 is a cartesian fibration of $\infty $-categories, and the evaluation functor

\[ \operatorname{ev}: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})\times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}\quad \quad (C, f_ C, u: C \rightarrow C') \mapsto f_ C(u) \]

exhibits $V$ as a cartesian conjugate of $U$.

Corollary 8.6.2.4. Every cocartesian fibration of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ admits a cartesian conjugate.

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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}', } \]

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. $\square$

Warning 8.6.2.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. If $\operatorname{\mathcal{C}}$ is not an $\infty $-category, then the morphism $V: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ given by Construction 8.6.2.2 need not be a cartesian conjugate of $U$. In ยง8.6.3, we will give an alternative construction of a cartesian conjugate which works in complete generality (Proposition 8.6.3.5).

The proof of Proposition 8.6.2.3 will require some preliminaries.

Lemma 8.6.2.6. Let $\lambda = (\lambda _{-}, \lambda _{+} ): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$ be a coupling of $\infty $-categories and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}_{+}$ be an isofibration. Then:

$(1)$

The projection map $\overline{V}: \operatorname{Fun}_{ / \operatorname{\mathcal{D}}^{+} }( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}$ is a cartesian fibration of $\infty $-categories.

$(2)$

Let $\widetilde{e}$ be a morphism in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}^{+} }( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$, corresponding to a morphism $e$ of $\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}$ and a functor $f_{e}: \Delta ^1 \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. Then $\widetilde{e}$ is $\overline{V}$-cartesian if and only if, for every morphism $u$ of $\Delta ^1 \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}$ whose image in $\operatorname{\mathcal{D}}_{+}$ is an isomorphism, the image $f_{e}(u)$ is an isomorphism in $\operatorname{\mathcal{E}}$.

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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{ /\operatorname{\mathcal{D}}_{+}}( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \ar [r] \ar [d]^{\overline{V}} & \operatorname{Fun}( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \ar [d]^{\overline{V}'} \\ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{D}}_{+} ), } \]

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 5.1.1.8). 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

\[ \xymatrix@R =50pt@C=50pt{ \{ D\} \times _{ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{C}}\ar [r]^-{ f_{D} } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \Delta ^1 \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}\ar@ {-->}[ur]^{f_{e}} \ar [r] & \operatorname{\mathcal{D}}_{+} } \]

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. $\square$

Lemma 8.6.2.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let $\overline{V}: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}})/ \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be the cartesian fibration of Lemma 8.6.2.6. Then:

$(1)$

Let $\widetilde{e}: (C, f_ C) \rightarrow (D, f_{D} )$ be a $\overline{V}$-cartesian morphism of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}})/ \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$. If $(D,f_ D)$ belongs to the simplicial subset $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ of Construction 8.6.2.2, then $(C, f_ C)$ also belongs to $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$.

$(2)$

The morphism $\overline{V}$ restricts to a cartesian fibration of $\infty $-categories

\[ V:\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}. \]
$(3)$

A morphism in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ is $V$-cartesian if and only if it is $\overline{V}$-cartesian when regarded as a morphism of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}})/ \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$.

Proof. We will prove assertion $(1)$; assertions $(2)$ and $(3)$ then follow as formal consequences (see Proposition 5.1.4.16). 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

\[ \xymatrix@R =50pt@C=50pt{ \widetilde{C} \ar [d]^{u} \ar [r] & \widetilde{D} \ar [d]^{v} \\ \widetilde{C}' \ar [r] & \widetilde{D}' } \]

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

\[ \xymatrix@R =50pt@C=50pt{ f_{C}(\widetilde{C}) \ar [d]^{f_{C}(u)} \ar [r] & f_{D}(\widetilde{D}) \ar [d]^{f_{D}(v)} \\ f_{C}(\widetilde{C}') \ar [r] & f_{D}(\widetilde{D}' )} \]

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 $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$. $\square$

We will deduce Proposition 8.6.2.3 from the following more precise result:

Proposition 8.6.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. Suppose we are given a commutative diagram

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

which we identify with a functor $F: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})$. The following conditions are equivalent:

$(a)$

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

$(b)$

The functor $F$ restricts to an equivalence of $\operatorname{\mathcal{E}}^{\dagger }$ with the full subcategory

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \]

introduced in Construction 8.6.2.2.

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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}^{\dagger } \ar [rr]^{F} \ar [dr]^{ U^{\dagger } } & & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \ar [dl]^{ V } \\ & \operatorname{\mathcal{C}}^{\operatorname{op}}, & } \]

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

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} \]

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

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

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 5.1.1.8. 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

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

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. $\square$

It follows from Proposition 8.6.2.8 that conjugate fibrations can be characterized by a universal mapping property:

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 $\operatorname{\mathcal{E}}$.

Example 8.6.2.10 (Functoriality). Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}\ar [rr]^{F} \ar [dr]^{U} & & \operatorname{\mathcal{E}}\ar [dl]_{V} \\ & \operatorname{\mathcal{C}}& } \]

where $U$ and $V$ are cocartesian fibrations, and the functor $F$ carries $U$-cocartesian morphisms of $\operatorname{\mathcal{D}}$ to $V$-cocartesian morphisms of $\operatorname{\mathcal{E}}$. Let $U^{\dagger }: \operatorname{\mathcal{D}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and $V^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be cartesian fibrations, and suppose we are given functors

\[ T_0: \operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}\quad \quad T: \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}} \]

which exhibit $U^{\dagger }$ and $V^{\dagger }$ as cartesian conjugates of $U$ and $V$, respectively. Then there is a functor $F^{\dagger }: \operatorname{\mathcal{D}}^{\dagger } \rightarrow \operatorname{\mathcal{E}}^{\dagger }$ satisfying $V^{\dagger } \circ F^{\dagger } = U^{\dagger }$ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T_0} \ar [d]^{F^{\dagger } \times \operatorname{id}} & \operatorname{\mathcal{D}}\ar [d]^{F} \\ \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} & \operatorname{\mathcal{E}}} \]

commutes up to isomorphism (in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})$). Moreover, the functor $F^{\dagger }$ is uniquely determined up to isomorphism (as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{D}}^{\dagger }, \operatorname{\mathcal{E}}^{\dagger }$).

Remark 8.6.2.11. In the situation of Example 8.6.2.10, the functor $F^{\dagger }$ automatically carries $U^{\dagger }$-cartesian morphisms of $\operatorname{\mathcal{D}}^{\dagger }$ to $V^{\dagger }$-cartesian morphisms of $\operatorname{\mathcal{E}}^{\dagger }$. See Remark 8.6.1.14.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. It follows from Proposition 8.6.2.8 that if there exists a functor

\[ T: \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}} \]

which exhibits $U^{\dagger }$ as a cartesian conjugate of $U$, then $U^{\dagger }$ can be recovered from $U$ up to equivalence. We close this section by showing that, in the same situation, we can also recover $U$ from the cartesian fibration $U^{\dagger }$.

Proposition 8.6.2.12. 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

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

which exhibits $U^{\dagger }$ as a cartesian conjugate of $U$. Then $T$ also 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 of Proposition 8.6.2.12. 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

\[ \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \xrightarrow {\operatorname{id}\times \lambda } \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} (\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{E}}^{\dagger } \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}, \]

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 6.2.2.22). Using Lemma 6.2.2.14, 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

\[ \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} \hookrightarrow \operatorname{\mathcal{E}}^{\dagger }_{0}(C) \xrightarrow { T^{C} } \operatorname{\mathcal{E}}_ C \]

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

\[ \xymatrix@C =50pt@R=50pt{ & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{w} & & Z } \]

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. $\square$