# Kerodon

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### 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 §8.6.4). Our main result can be stated as follows:

Proposition 8.6.6.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets. Then $U$ is a cocartesian dual of $U^{\vee }$ if and only if the opposite fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cartesian conjugate of $U^{\vee }$.

Before giving the proof of Proposition 8.6.6.1, let us collect some consequences.

Corollary 8.6.6.2 (Symmetry). 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 $U^{\dagger , \operatorname{op}}$.

Corollary 8.6.6.3. For every simplicial set $\operatorname{\mathcal{C}}$, the formation of cartesian conjugates induces a bijection

$\xymatrix@C =50pt@R=50pt{ \{ \textnormal{Cocartesian fibrations U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}} \} / \textnormal{Equivalence} \ar [d] \\ \{ \textnormal{Cartesian fibrations U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}} \} / \textnormal{Equivalence.} }$

Proof. Combine Proposition 8.6.6.1 with Corollary 8.6.5.2. $\square$

Example 8.6.6.4. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration, which is cocartesian dual to the projection map $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ of Notation 8.1.1.6. This follows by combining Proposition 8.6.6.1 with Corollary 8.6.3.18 (applied to the opposite $\infty$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$).

Corollary 8.6.6.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $L$ be the collection of $U$-cocartesian edges of $\operatorname{\mathcal{E}}$, and let $\operatorname{\mathcal{E}}^{\vee }$ be the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{Cospan}(\operatorname{\mathcal{C}}) } \operatorname{Cospan}^{L, \mathrm{all}}( \operatorname{\mathcal{E}})$. Then the projection map $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration, which is a cocartesian dual of $U$.

Remark 8.6.6.6. 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 8.6.5.1 and Corollary 8.6.3.14, $U$ 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

$\operatorname{Cospan}^{ \mathrm{all}, L \cap R }(\operatorname{\mathcal{E}}) = \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}}).$
• Using Variant 8.1.7.14, we can identify $U^{\operatorname{op}}$ with the map

$\operatorname{Cospan}^{ L \cap R , \mathrm{all}}(\operatorname{\mathcal{E}}) = \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}}).$
• Using Proposition 8.6.3.5 (and Remark 8.6.3.7), we can identify $U^{\dagger }$ with the projection map $\operatorname{Cospan}^{R,L}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{ \mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}})$.

• Using Corollary 8.6.6.5 (and Remark 8.6.3.7), we can identify $U^{\vee }$ with the map $\operatorname{Cospan}^{L, R}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$.

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 $U$. $\square$

We now turn to the proof of Proposition 8.6.6.1. Consider first the special case where $\operatorname{\mathcal{C}}= \Delta ^0$. For any $\infty$-category $\operatorname{\mathcal{E}}$, the projection map $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^0$ can be regarded as a cartesian conjugate of itself (Example 8.6.1.3). Consequently, Proposition 8.6.6.1 reduces to the assertion that $U$ is dual to the opposite fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \Delta ^0$: that is, that there exists a balanced profunctor $\mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}$. This is proved by taking $\mathscr {H}$ to be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{E}}$, in the sense of Definition 8.3.3.1. To prove Proposition 8.6.6.1, we will need a relative variant of this construction.

Definition 8.6.6.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets, so that the induced map

$\lambda : \operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$

is a left fibration (Proposition 8.1.1.15). Let $\kappa$ be an uncountable cardinal. We say that a diagram

$\mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$

is a relative $\operatorname{Hom}$-functor for $U$ if it is a covariant transport representation for the left fibration $\lambda$ (Definition 5.6.5.1).

Remark 8.6.6.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let

$\lambda : \operatorname{Tw}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$

be the left fibration of Proposition 8.1.1.15. If $X$ and $Y$ are vertices of $\operatorname{\mathcal{E}}$ and $f: U(X) \rightarrow U(Y)$ is an edge of $\operatorname{\mathcal{C}}$, then the fiber $\lambda ^{-1} \{ (X,f,Y) \}$ is a Kan complex, which is homotopy equivalent to the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{E}}_{f} }( X, Y )$, where we abuse notation by identifying $X$ and $Y$ with their preimages in the $\infty$-category $\operatorname{\mathcal{E}}_{f} = \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Consequently, if $\mathscr {H}$ is a relative $\operatorname{Hom}$-functor for $U$, then we have homotopy equivalences $\mathscr {H}(X,f,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{E}}_{f}}(X,Y)$.

Remark 8.6.6.10 (Existence and Uniqueness). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $\kappa$ be an uncountable cardinal. Then $U$ admits a relative $\operatorname{Hom}$-functor

$\mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$

if and only if the inner fibration $U$ is locally $\kappa$-small (see Proposition 4.7.9.5). Moreover, if this condition is satisfied, then $\mathscr {H}$ is unique up to isomorphism (Corollary 5.6.0.6).

Example 8.6.6.11. Let $\operatorname{\mathcal{E}}$ be a locally $\kappa$-small $\infty$-category and let $\mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{E}}$. Suppose we are given a functor $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(J)$ is the nerve of a partially ordered set $J$. Then $U$ is automatically an inner fibration (Proposition 4.1.1.10). Moreover, the iterated fiber product $\operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ can be identified with the full subcategory of $\operatorname{\mathcal{E}}^{\operatorname{op}} \times \operatorname{\mathcal{E}}$ spanned by those objects $(X,Y)$ satisfying $U(X) \leq U(Y)$ (see Example 8.1.0.5). In this case, the restriction of $\mathscr {H}$ to this full subcategory is a relative $\operatorname{Hom}$-functor for the inner fibration $U$, in the sense of Definition 8.6.6.8.

Remark 8.6.6.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty$-categories and let

$\mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$

be a relative $\operatorname{Hom}$-functor for $U$. Then, for every vertex $C \in \operatorname{\mathcal{C}}$, the restriction of $\mathscr {H}$ to

$\operatorname{\mathcal{E}}_{C}^{\operatorname{op}} \times \operatorname{\mathcal{E}}_{C} \simeq \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ \operatorname{id}_ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$

is $\operatorname{Hom}$-functor for the $\infty$-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, in the sense of Definition 8.3.3.1.

Notation 8.6.6.13 (Relative Yoneda Embeddings). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets, let $\kappa$ be an uncountable cardinal, and let

$\mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$

be a relative $\operatorname{Hom}$-functor for $U$. Then $\mathscr {H}$ can be identified with a morphism

$T: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ),$

where $\operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$ denotes the relative exponential of Construction 4.5.9.1. In this case, we will refer to $T$ as the relative Yoneda embedding of the inner fibration $U$.

Remark 8.6.6.14. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. Then the relative Yoneda embedding $T$ of Notation 8.6.6.13 fits into a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} & \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \ar [d] \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}. }$

Consequently, for each vertex $C \in \operatorname{\mathcal{C}}$, the restriction of $T$ to $\operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ \operatorname{id}_{C} \}$ can be regarded as a functor

$T_{C}: \operatorname{\mathcal{E}}_{C}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{S}}^{< \kappa } ),$

which is a contravariant Yoneda embedding for the $\infty$-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, in the sense of Definition 8.3.3.9. See Remark 8.6.6.12.

We now formulate a more precise version of Proposition 8.6.6.1.

Theorem 8.6.6.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $\kappa$ be an uncountable cardinal for which $U$ is locally $\kappa$-small, and let

$T: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{ < \kappa } )$

be a relative Yoneda embedding for $U$ (see Notation 8.6.6.13). Then:

$(a)$

The morphism $T$ factors through the simplicial subset $\operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ of Construction 8.6.5.6.

$(b)$

The morphism $T$ exhibits $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ as a cartesian conjugate of the cocartesian fibration $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{C}}$.

Proof. We first prove $(a)$. Fix an edge $e: C_0 \rightarrow C_1$ of $\operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{E}}_{0}$ denote the fiber $\{ C_0 \} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, and define $\operatorname{\mathcal{E}}_{1}$ similarly. Fix an object $Y \in \operatorname{\mathcal{E}}_0$, so that we can identify the pair $(Y,e)$ with a vertex of the fiber product $\operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. Then $T(Y,e)$ can be regarded as a functor $F: \operatorname{\mathcal{E}}_{1} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, and we wish to show that $F$ is corepresentable. Replacing $U$ by the projection map $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^1$, we can reduce to the special case where $\operatorname{\mathcal{C}}= \Delta ^1$ is the standard $1$-simplex (with $C_0 = 0$ and $C_1 = 1$). Our assumption that $U$ is a cocartesian fibration guarantees that there exists a $U$-cocartesian morphism $f: Y \rightarrow X$ with $X \in \operatorname{\mathcal{E}}_1$. Let $\mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{E}}$. Using Example 8.6.6.11, we can identify $F$ with the functor

$\operatorname{\mathcal{E}}_{1} \hookrightarrow \operatorname{\mathcal{E}}\xrightarrow { \mathscr {H}(Y,-) } \operatorname{\mathcal{S}}^{< \kappa },$

so that $f$ can be identified with a vertex of $\eta \in F(X)$. Since $f$ is $U$-cocartesian, the vertex $\eta$ exhibits $F$ as corepresented by the object $X$ (see Lemma 8.6.5.14).

We now prove $(b)$. For each vertex $C \in \operatorname{\mathcal{C}}$, the restriction of $T$ to the fiber $\operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ \operatorname{id}_{C} \}$ determines a functor $T_{C}: \operatorname{\mathcal{E}}^{\operatorname{op}}_{C} \rightarrow \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{S}}^{< \kappa } )$ which is a contravariant Yoneda embedding for the $\infty$-category $\operatorname{\mathcal{E}}_{C}$ (Remark 8.6.6.14), and is therefore an equivalence of $\infty$-categories (Theorem 8.3.3.13). We will complete the proof by showing that $T$ satisfies conditions $(2')$ and $(2'')$ of Proposition 8.6.1.13. Both of these conditions make an assertion about an arbitrary edge $e: C_0 \rightarrow C_1$ of the simplicial set $\operatorname{\mathcal{C}}$; let us denote these assertions by $(2'_ e)$ and $(2''_ e)$, respectively. To verify them, we can replace $U$ by the projection map $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ as above, and thereby reduce to the special case where $\operatorname{\mathcal{C}}= \Delta ^1$ is the standard $1$-simplex and $e$ is the nondegenerate edge of $\operatorname{\mathcal{C}}$. Let $\mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{E}}$. Using Example 8.6.6.11 again and Variant 8.6.5.11, we can formulate the hypotheses of Proposition 8.6.1.13 more concretely as follows:

$(2'_ e)$

For every object $Y \in \operatorname{\mathcal{E}}_0$, the functor $\mathscr {H}(Y, -): \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ is left Kan extended from $\operatorname{\mathcal{E}}_0$.

$(2''_{e})$

For every object $Y \in \operatorname{\mathcal{E}}_0$ and every $U$-cocartesian morphism $f: Y \rightarrow X$ with $X \in \operatorname{\mathcal{E}}_1$, the induced natural transformation $\mathscr {H}(X, -) \rightarrow \mathscr {H}(Y,-)$ is an isomorphism when restricted to the subcategory $\operatorname{\mathcal{E}}_1$. In other words, for every object $W \in \operatorname{\mathcal{E}}_1$, precomposition with $f$ induces a homotopy equivalence $\mathscr {H}(X,W) \rightarrow \mathscr {H}(Y,W)$.

Assertion $(2'_ e)$ now follows from Lemma 8.6.5.13, and assertion $(2''_{e})$ from Corollary 5.1.2.3. $\square$

Proof of Proposition 8.6.6.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets. Fix an uncountable cardinal $\kappa$ such that $U$ and $U^{\vee }$ are locally $\kappa$-small. If $U$ is a cocartesian dual of $U^{\vee }$, then Proposition 8.6.5.18 guarantees that $U^{\vee }$ is equivalent to the cocartesian fibration $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{C}}$. Combining this observation with Theorem 8.6.6.15, we conclude that $U^{\operatorname{op}}$ is a cartesian conjugate of $U^{\vee }$.

We now prove the converse. Assume that $U^{\operatorname{op}}$ is a cartesian conjugate of $U^{\vee }$; we wish to show that $U$ is a cartesian dual of $U^{\vee }$. Applying Theorem 8.6.5.1, we see that $U^{\vee }$ admits a cocartesian dual $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$. The first part of the proof shows that the opposite fibration $U'^{\operatorname{op}}: \operatorname{\mathcal{E}}'^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cartesian conjugate of $U^{\vee }$, and is therefore equivalent to the cartesian fibration $U^{\operatorname{op}}$ (Corollary 8.6.3.14). It follows that the cocartesian fibrations $U$ and $U'$ are also equivalent, so that $U$ is also a cocartesian dual of $U^{\vee }$. $\square$