Proposition 8.6.6.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set, 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. Then $U^{\dagger }$ is a cartesian conjugate of $U$ (in the sense of Definition 8.6.1.1) if and only if the opposite fibration $U^{\dagger , \operatorname{op}}: \operatorname{\mathcal{E}}^{\dagger ,\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian dual of $U$ (in the sense of Definition 8.6.3.1).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
8.6.6 Comparison of Dual and Conjugate Fibrations
In this section, we show that the theory of conjugate fibrations (introduced in §8.6.1) can be regarded as a reformulation of cocartesian duality (introduced in §8.6.3). Our main result can be stated as follows:
Corollary 8.6.6.2. Let $\operatorname{\mathcal{C}}$ be a simplicial set, 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. Then $U^{\dagger }$ is a cartesian conjugate of $U$ if and only if $U^{\operatorname{op}}$ is a cartesian conjugate of $U^{\dagger , \operatorname{op}}$.
Proof. Combine Proposition 8.6.6.1 with Remark 8.6.3.3. $\square$
Corollary 8.6.6.3. 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 The following conditions are equivalent:
The functor $T$ exhibits $U^{\dagger }$ as a cartesian conjugate of $U$ (in the sense of Definition 8.6.1.1).
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.
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{ U } \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}. } \]
Proof. We will show that $(2)$ implies $(1)$; the reverse implication follows from Proposition 8.6.2.11. Using Corollary 8.6.6.2 and Corollary 8.6.2.4, 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 8.6.2.11 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 $U$. $\square$
Our proof of Proposition 8.6.6.1 will require some preliminaries.
Construction 8.6.6.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set, and suppose we are given a pair of morphisms $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$. Let $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ be a morphism of simplicial sets for which the diagram is commutative. Let $\lambda _{+}: \operatorname{Tw}( \operatorname{\mathcal{E}}^{\dagger } ) \rightarrow \operatorname{\mathcal{E}}^{\dagger }$ be the projection map of Notation 8.1.1.6 and let $\iota : \operatorname{Tw}(\operatorname{\mathcal{C}}^{\operatorname{op}}) \xrightarrow {\sim } \operatorname{Tw}(\operatorname{\mathcal{C}})$ be the isomorphism described in Remark 8.1.1.7. Then we can extend (8.81) to a commutative diagram Using Proposition 8.1.3.7, we can identify the outer rectangle with a diagram where the lower horizontal map is the monomorphism of Variant 8.1.7.14. Passing to opposite simplicial sets (and invoking Remark 8.1.3.4), we obtain a comparison map where $\operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ is the simplicial set defined in Notation 8.6.5.1.
8.81
\begin{equation} \begin{gathered}\label{equation:compare-dagger-with-conjugacy} \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}}} \end{gathered} \end{equation}
8.82
\begin{equation} \begin{gathered}\label{equation:compare-dagger-with-conjugacy2} \xymatrix@C =50pt@R=50pt{ \operatorname{Tw}(\operatorname{\mathcal{E}}^{\dagger }) \ar [r]^-{(\lambda _{+}, \iota \circ \operatorname{Tw}(U^{\dagger }))} \ar [d]^{ \operatorname{Tw}(U^{\dagger }) } & \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}}^{\operatorname{op}} ) \ar [r]^-{ \iota } & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
8.83
\begin{equation} \begin{gathered}\label{equation:compare-dagger-with-conjugacy3} \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \ar [r] \ar [d]^{U^{\dagger }} & \operatorname{Cospan}( \operatorname{\mathcal{E}}) \ar [d]^{ \operatorname{Cospan}(U) } \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \ar [r] & \operatorname{Cospan}( \operatorname{\mathcal{C}}), } \end{gathered} \end{equation}
\[ \Psi : \operatorname{\mathcal{E}}^{\dagger , \operatorname{op}} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Cospan}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}(\operatorname{\mathcal{E}}) = \operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}), \]
Remark 8.6.6.5. In the situation of Construction 8.6.6.4, the comparison map can be described explicitly on low-dimensional simplices as follows:
If $X$ is a vertex of $\operatorname{\mathcal{E}}^{\dagger }$ having image $C = U^{\dagger }(X)$, then $\Psi (X)$ is the vertex of $\operatorname{Cospan}( \operatorname{\mathcal{E}})$ corresponding to the vertex $T( X, \operatorname{id}_{ C } ) \in \operatorname{\mathcal{E}}$.
Let $X$ and $Y$ be vertices of $\operatorname{\mathcal{E}}^{\dagger }$, having images $C= U^{\dagger }(X)$ and $D = U^{\dagger }(Y)$. Let $f: Y \rightarrow X$ be an edge of $\operatorname{\mathcal{E}}^{\dagger }$, and let us identify $U^{\dagger }(f)$ with an edge $e: C \rightarrow D$ in the simplicial set $\operatorname{\mathcal{C}}$. Then $\Psi ( f): \Psi ( X) \rightarrow \Psi ( Y)$ is the edge of $\operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ corresponding to the pair of edges $T(X, \operatorname{id}_{C} ) \xrightarrow { T( \operatorname{id}_{X} , e_{L} ) } T( X, e ) \xleftarrow { T(f, e_{R} ) } T(Y, \operatorname{id}_{D} )$ in $\operatorname{\mathcal{E}}$; here $e_{L}: \operatorname{id}_{C} \rightarrow e$ and $e_{R}: \operatorname{id}_{D} \rightarrow e$ denote the edges of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ described in Example 8.1.3.6.
\[ \Psi : \operatorname{\mathcal{E}}^{\dagger , \operatorname{op}} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{E}}) \]
We will deduce Proposition 8.6.6.1 from the following more precise result:
Proposition 8.6.6.6. Let $\operatorname{\mathcal{C}}$ be a simplicial set, 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 morphism $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ for which the diagram is commutative. The following conditions are equivalent:
The morphism $T$ exhibits $U^{\dagger }$ as a cartesian conjugate of $U_{+}$, in the sense of Definition 8.6.1.1.
The comparison map $\Psi : \operatorname{\mathcal{E}}^{\dagger , \operatorname{op}} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ of Construction 8.6.6.4 factors through the simplicial subset $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ of Notation 8.6.5.1. Moreover, $\Psi $ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$.
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{ U } \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}} \]
Proof. Using Proposition 5.1.7.14, we see that $(b)$ is equivalent to the following three conditions:
- $(b_0)$
The map $\Psi $ factors through the simplicial subset $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$.
- $(b_1)$
Let $U^{\vee }: \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the cocartesian fibration of Lemma 8.6.5.10. Then $\Psi $ carries $U^{\dagger }$-cartesian edges of $\operatorname{\mathcal{E}}^{\dagger }$ to $U^{\vee }$-cocartesian edges of $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$.
- $(b_2)$
For each vertex $C \in \operatorname{\mathcal{C}}$, the morphism $\Psi $ restricts to an equivalence of $\infty $-categories
\[ \Psi _{C}: (\operatorname{\mathcal{E}}^{\dagger }_{C})^{\operatorname{op}} \rightarrow \{ C\} \times _{ \operatorname{\mathcal{C}}} \operatorname{Cospan}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) = \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{E}}_{C} ). \]
For every edge $e: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, let $e_{L}: \operatorname{id}_{C} \rightarrow e$ and $e_{R}: \operatorname{id}_{D} \rightarrow e$ denote the edges of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ described in Example 8.1.3.6. Using Remark 8.6.6.5, we can rewrite condition $(b_0)$ as follows:
- $(b'_0)$
Let $X$ be a vertex of $\operatorname{\mathcal{E}}^{\dagger }$ having image $C = U^{\dagger }(X)$ in $\operatorname{\mathcal{C}}$, and let $e: C \rightarrow D$ be an edge of $\operatorname{\mathcal{C}}$. Then $T( \operatorname{id}_{X}, e_{L} ): T(X, \operatorname{id}_{C} ) \rightarrow T( X, e )$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
Similarly, by combining Remark 8.6.6.5 with the characterization of $U^{\vee }$-cartesian edges supplied by Lemma 8.6.5.10, we can rewrite condition $(b_1)$ as follows:
- $(b'_1)$
Let $f: Y \rightarrow X$ be a $U^{\dagger }$-cartesian edge of $\operatorname{\mathcal{E}}^{\dagger }$, and let us identify $U^{\dagger }(f)$ with an edge $e: C \rightarrow D$ of $\operatorname{\mathcal{C}}$. Then $T(f, e_{R} ): T(Y, \operatorname{id}_{D}) \rightarrow T(X,e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$.
Unwinding the definitions, we observe that for each vertex $C \in \operatorname{\mathcal{C}}$, the functor $\Psi _{C}$ factors as a composition
\[ (\operatorname{\mathcal{E}}^{\dagger }_{C})^{\operatorname{op}} \xrightarrow { T_{C}^{\operatorname{op}} } \operatorname{\mathcal{E}}^{\operatorname{op}}_{C} \hookrightarrow \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{E}}_{C} ), \]
where the second map is the equivalence of Variant 8.1.7.14. We can therefore rewrite $(b_2)$ as follows:
- $(b'_2)$
For each vertex $C \in \operatorname{\mathcal{C}}$, the morphism $T$ restricts to an equivalence of $\infty $-categories $T_{C}: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$.
The equivalence of $(a)$ and $(b)$ now follows from Proposition 8.6.1.13. $\square$
Corollary 8.6.6.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then the projection map $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cartesian conjugate of $U$.
Proof. Using Corollary 8.6.2.4, we can choose a cartesian fibration $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and a morphism $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ which exhibits $U^{\dagger }$ as a cartesian conjugate of $U$. Applying Proposition 8.6.6.6, we see that the comparison map of Construction 8.6.6.4 provides a morphism $\operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})^{\operatorname{op}}$ which is an equivalence of cartesian fibrations over $\operatorname{\mathcal{C}}^{\operatorname{op}}$. It follows that the projection map $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is also a cartesian conjugate of $U$. $\square$
Corollary 8.6.6.8 (Uniqueness). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then $U$ admits a cartesian conjugate, which is uniquely determined up equivalence.
Proof. Combining Proposition 8.6.6.6 with Corollary 8.6.6.7, we see that a cartesian fibration $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is conjugate to $U$ if and only if it is equivalent to the projection map $\operatorname{Cospan}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$. $\square$
Example 8.6.6.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Applying Construction 8.6.6.4 to the evaluation functor we obtain a comparison map which is an equivalence of $\infty $-categories (Proposition 8.6.6.6).
\[ \operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}\quad \quad (C, f_ C, u: C \rightarrow C') \mapsto f_ C(u), \]
\[ \Psi : \operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})^{\operatorname{op}} \rightarrow \operatorname{Cospan}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}) \]
Warning 8.6.6.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Corollary 8.6.6.7 guarantees that existence of a morphism $T: \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ which exhibits the projection map $\operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ as a cartesian conjugate of $U$. Beware that the construction of $T$ requires making some auxiliary choices. For example, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then we can construct the datum $T$ by choosing a homotopy inverse to the equivalence $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}})^{\operatorname{op}} \rightarrow \operatorname{Cospan}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ of Example 8.6.6.9.
Proof of Proposition 8.6.6.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration. Let $U^{\vee }: \operatorname{Cospan}^{\operatorname{CCart}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the projection map. Then $U^{\vee }$ is a cocartesian dual of $U$ (Theorem 8.6.5.6), and the opposite fibration $U^{\vee , \operatorname{op}}$ is a cartesian conjugate of $U$ (Corollary 8.6.6.7). Let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration of simplicial sets. Using the uniqueness assertions of Theorem 8.6.4.1 and Corollary 8.6.6.8, we see that the following conditions are equivalent:
The fibration $U^{\dagger }$ is a cartesian conjugate of $U$.
The fibration $U^{\dagger }$ is equivalent to $U^{\vee , \operatorname{op}}$ (as a cartesian fibration over $\operatorname{\mathcal{C}}^{\operatorname{op}}$).
The fibration $U^{\dagger , \operatorname{op}}$ is equivalent to $U^{\vee }$ (as a cocartesian fibration over $\operatorname{\mathcal{C}}$).
The fibration $U^{\dagger , \operatorname{op}}$ is a cocartesian dual of $U$.
Remark 8.6.6.11. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and let $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be the opposite fibration. By virtue of Theorem 8.6.4.1 and Corollary 8.6.6.8, $U$ admits a cocartesian dual $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ and a cartesian conjugate $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$, which are uniquely determined up to equivalence and opposite to one another (Proposition 8.6.6.1). When $\operatorname{\mathcal{C}}$ is an $\infty $-category, all four of these fibrations can be realized as a suitable restriction of the projection map $\operatorname{Cospan}(U): \operatorname{Cospan}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. Let $L$ denote the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$, and let $R$ denote the collection of all morphisms $f$ of $\operatorname{\mathcal{E}}$ such that $U(f)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then:
Using Proposition 8.1.7.6, we can identify $U$ with the map
Using Variant 8.1.7.14, we can identify $U^{\operatorname{op}}$ with the map
Using Theorem 8.6.5.6 (and Remark 8.6.5.9), we can identify $U^{\vee }$ with the map $\operatorname{Cospan}^{L, R}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$.
Using Proposition 8.6.6.6 (and Remark 8.6.5.9), we can identify $U^{\dagger }$ with the map $\operatorname{Cospan}^{R,L}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}})$.
\[ \operatorname{Cospan}^{ \mathrm{all}, L \cap R }(\operatorname{\mathcal{E}}) = \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}). \]
\[ \operatorname{Cospan}^{ L \cap R , \mathrm{all}}(\operatorname{\mathcal{E}}) = \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}}). \]