Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

7.3.2 Digression: Direct Image Fibrations

Suppose we are given morphisms of simplicial sets $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, where $V$ is an inner fibration. For every vertex $C \in \operatorname{\mathcal{C}}$, set $\operatorname{\mathcal{D}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, so that $V$ restricts to an inner fibration $V_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{D}}_{C}$. Sections of the inner fibration $V_{C}$ are parametrized by the simplicial set

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{C} }( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} ) = \operatorname{Fun}( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} ) \times _{ \operatorname{Fun}( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{D}}_{C} ) } \{ \operatorname{id}_{ \operatorname{\mathcal{D}}_{C} } \} , \]

which is an $\infty $-category (Proposition 4.1.4.6). Our goal in this section is to study the dependence of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{C} }( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} )$ on the vertex $C \in \operatorname{\mathcal{C}}$. We begin by observing that $\operatorname{Fun}_{/ \operatorname{\mathcal{D}}_{C} }( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} )$ can be identified with the the fiber over $C$ of a morphism $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$.

Construction 7.3.2.1 (Direct Images). Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets. For every integer $n \geq 0$, we let $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{n}$ denote the collection of pairs $(\sigma , f)$, where $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}$ and $f: \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a morphism for which the composition

\[ \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}\xrightarrow {V} \operatorname{\mathcal{D}} \]

coincides with projection onto the second factor. Note that every nondecreasing function $\alpha : [m] \rightarrow [n]$ induces a map

\[ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{n} \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{m} \quad \quad (\sigma ,f) \mapsto (\alpha ^{\ast }(\sigma ), f' ), \]

where $f'$ denotes the composite map

\[ \Delta ^{m} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow { \alpha \times \operatorname{id}} \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}. \]

This construction is compatible with composition, and therefore endows $\{ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})_{n} \} _{n \geq 0}$ with the structure of a simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) = \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{\bullet }$ which we will refer to as the direct image of $\operatorname{\mathcal{E}}$ along $U$.

Note that the construction $(\sigma ,f) \mapsto \sigma $ determines a morphism of simplicial sets $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})$. Moreover, there is a tautological evaluation map

\[ \operatorname{ev}: \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}, \]

which carries an $n$-simplex $( \widetilde{\sigma }, (\sigma ,f) )$ of the fiber product $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ to the $n$-simplex of $\operatorname{\mathcal{E}}$ given by the composite map

\[ \Delta ^{n} \xrightarrow { \operatorname{id}\times \widetilde{\sigma } } \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}. \]

The direct image $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ of Construction 7.3.2.1 is characterized by a universal property:

Proposition 7.3.2.2. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets. For every morphism of simplicial sets $\sigma : \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, postcomposition with the evaluation map $\operatorname{ev}: \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$ of Construction 7.3.2.1 induces a bijection

\[ \operatorname{Hom}_{ (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}} }( \operatorname{\mathcal{C}}', \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Hom}_{ (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}} }( \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}). \]

Proof. Writing $\operatorname{\mathcal{C}}'$ as a colimit of simplices, we may reduce to the case where $\operatorname{\mathcal{C}}' = \Delta ^{n}$, so that $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}$. In this case, the desired result follows immediately from the definition of the simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$. $\square$

Remark 7.3.2.3. In the situation of Proposition 7.3.2.2, precomposition with the evaluation map $\operatorname{ev}: \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$ induces an isomorphism of simplicial sets

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) ) \xrightarrow {\sim } \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}). \]

The bijectivity of this map on $n$-simplices follows by applying Proposition 7.3.2.2 after replacing $\operatorname{\mathcal{C}}'$ by the product $\Delta ^ n \times \operatorname{\mathcal{C}}'$.

Corollary 7.3.2.4. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the pullback functor

\[ U^{\ast }: (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}} \rightarrow (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}} \quad \quad \operatorname{\mathcal{C}}' \mapsto \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}} \]

has a right adjoint, given on objects by the construction $\operatorname{\mathcal{E}}\mapsto \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$.

Example 7.3.2.5. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ be the identity map. Then the projection map $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ is an isomorphism.

Example 7.3.2.6. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, and let $U: \operatorname{\mathcal{D}}\rightarrow \Delta ^{0}$ denote the projection map. Then the direct image $\operatorname{Res}_{ \operatorname{\mathcal{D}}/ \Delta ^{0} }( \operatorname{\mathcal{E}})$ can be identified with the simplicial set

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{C} }( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} ) = \operatorname{Fun}( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} ) \times _{ \operatorname{Fun}( \operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{D}}_{C} ) } \{ \operatorname{id}_{ \operatorname{\mathcal{D}}_{C} } \} , \]

which parametrizes sections of $V$.

Remark 7.3.2.7. Suppose we are given a commutative diagram of simplicial sets

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

where both squares are pullbacks. Then there is a canonical isomorphism of simplicial sets

\[ \operatorname{Res}_{ \operatorname{\mathcal{D}}' / \operatorname{\mathcal{C}}' }( \operatorname{\mathcal{E}}' ) \simeq \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}). \]

Remark 7.3.2.8. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets, and let $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ be the projection map of Construction 7.3.2.1. For every vertex $C \in \operatorname{\mathcal{C}}$, Remark 7.3.2.7 and Example 7.3.2.6 furnish an isomorphism of simplicial sets

\[ \pi ^{-1} \{ C\} = \{ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{C} }(\operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} ). \]

Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets. It follows from Remark 7.3.2.8 and Proposition 4.1.4.6 that, for any morphism of simplicial sets $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, the fibers of the induced map $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ are $\infty $-categories.

Exercise 7.3.2.9. Let $\operatorname{\mathcal{C}}= \Delta ^2$ be the standard $2$-simplex, let $\operatorname{\mathcal{D}}= \operatorname{N}_{\bullet }( \{ 0 < 2 \} )$ be the long edge of $\Delta ^2$, and let $\operatorname{\mathcal{C}}= \{ 0\} \coprod \{ 2\} $ be its boundary. Let $V: \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}$ be the inclusion maps. Show that $U$ are $V$ are isofibrations of $\infty $-categories but that the projection map $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ can be identified with the horn inclusion $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$, which is not an inner fibration.

To avoid the behavior described in Exercise 7.3.2.9, we need to impose an additional condition on the morphism $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

Definition 7.3.2.10. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. We will say that $U$ is exponentiable if it satisfies the following condition:

$(\ast )$

For every diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}'' \ar [r]^-{F} \ar [d] & \operatorname{\mathcal{D}}' \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}'' \ar [r]^-{ \overline{F} } & \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}} \]

in which both squares are pullbacks, if $\overline{F}$ is a categorical equivalence, then $F$ is also a categorical equivalence.

Remark 7.3.2.11. We will be primarily interested in the special case of Definition 7.3.2.10 where $U$ is an inner fibration of simplicial sets. In this case, Definition 7.3.2.10 can be considerably simplified: to show that an inner fibration of simplicial sets $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is exponentiable, it suffices to verify condition $(\ast )$ in the special case where $\overline{F}: \operatorname{\mathcal{C}}'' \rightarrow \operatorname{\mathcal{C}}'$ is the inner horn $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$ (see Proposition ).

Remark 7.3.2.12. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be exponentiable morphisms of simplicial sets. Then the composition $(U \circ V): \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is also exponentiable.

Remark 7.3.2.13. Suppose we are given a pullback diagram of simplicial sets

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

If $U$ is exponentiable, then $U'$ is also exponentiable.

Remark 7.3.2.14. The collection of exponentiable morphisms of simplicial sets is closed under retracts. That is, if we are given a commutative diagram of simplicial sets

\[ \xymatrix@C =40pt@R=40pt{ \operatorname{\mathcal{D}}\ar [r] \ar [d]^{U} & \operatorname{\mathcal{D}}' \ar [d]^{U'} \ar [r] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}} \]

where $U'$ is exponentiable and both horizontal compositions are the identity, then $U$ is also exponentiable.

Example 7.3.2.15. Let $\operatorname{\mathcal{D}}$ be any simplicial set. Then the projection map $\operatorname{\mathcal{D}}\rightarrow \Delta ^{0}$ is exponentiable (this is a reformulation of Remark 4.5.2.7).

Example 7.3.2.16. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets which is either a cartesian fibration or a cocartesian fibration. Then $U$ is exponentiable (this is a reformulation of Corollary 5.6.4.6).

Example 7.3.2.17. The inclusion map $\operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \hookrightarrow \Delta ^2$ is an isofibration of $\infty $-categories which is not exponentiable. Note that there is a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \{ 0 \} \coprod \{ 2\} \ar [r] \ar [d] & \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \ar [d] \\ \Lambda ^{2}_{1} \ar [r] & \Delta ^2 } \]

where the lower horizontal map is a categorical equivalence, but the upper horizontal map is not.

The terminology of Definition 7.3.2.10 is motivated by the following:

Proposition 7.3.2.18. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $U$ is exponentiable (Definition 7.3.2.10).

$(2)$

Let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be a simplicial subset for which the restriction $U|_{\operatorname{\mathcal{D}}_0}$ is exponentiable, let $T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ be an isofibration in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}}$, and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{D}}_0 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}'_0 = \operatorname{\mathcal{D}}_0 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}'$. Then the induced map

\[ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Res}_{ \operatorname{\mathcal{D}}_0/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}_0 ) \times _{ \operatorname{Res}_{\operatorname{\mathcal{D}}_0/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}'_0) } \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}' ) \]

is also an isofibration.

$(3)$

For every isofibration $T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}}$, the induced map

\[ \operatorname{Res}_{ \operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}') \]

is also an isofibration.

$(4)$

For every isofibration of $\infty $-categories $T_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}'_0$, the induced map

\[ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}_0 ) \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}'_0 ) \]

is also an isofibration.

Proof. We first show that $(1)$ implies $(2)$. Assume that $U$ is exponentiable, let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be a simplicial subset for which $U|_{\operatorname{\mathcal{D}}_0}$ is also exponentiable, let $T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ be an isofibration in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}}$, and let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets which is a categorical equivalence; we wish to show that every lifting problem

7.14
\begin{equation} \begin{gathered}\label{equation:exponentiable-equivalent-condition} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \ar [d] \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{Res}_{ \operatorname{\mathcal{D}}_0/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}_0 ) \times _{ \operatorname{Res}_{\operatorname{\mathcal{D}}_0/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}'_0) } \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}' } \end{gathered} \end{equation}

admits a solution. Note that the bottom horizontal map determines a morphism of simplicial sets $B \rightarrow \operatorname{\mathcal{C}}$. Invoking the universal property of direct images (Proposition 7.3.2.2), we can rewrite (7.14) as a lifting problem

7.15
\begin{equation} \begin{gathered}\label{equation:silly-diagram-for-fun2} \xymatrix@R =50pt@C=50pt{ (A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}) \coprod _{ (A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0)} (B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0 ) \ar [d]^{j} \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{T} \\ B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{E}}'. } \end{gathered} \end{equation}

Since $U$ and $U|_{\operatorname{\mathcal{D}}_0}$ are exponentiable, the horizontal maps in the diagram

7.16
\begin{equation} \begin{gathered}\label{equation:silly-diagram-for-fun} \xymatrix@R =50pt@C=50pt{ A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0 \ar [r] \ar [d] & B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0 \ar [d] \\ A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\ar [r] & B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}} \end{gathered} \end{equation}

are categorical equivalences. In particular, the diagram (7.16) is a categorical pushout square (Proposition 4.5.3.7). It follows that the morphism $j$ appearing in (7.15) is also a categorical equivalence (Proposition 4.5.3.8). Since $T$ is an isofibration of simplicial sets, it follows that the lifting problem (7.15) admits a solution.

The implication $(2) \Rightarrow (3)$ and $(3) \Rightarrow (4)$ are immediate. We will complete the proof by showing that $(4)$ implies $(1)$. Assume that condition $(4)$ is satisfied and suppose that we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}'' \ar [r]^-{F} \ar [d] & \operatorname{\mathcal{D}}' \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}'' \ar [r]^-{ \overline{F} } & \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}} \]

where both squares are pullbacks and $\overline{F}$ is a categorical equivalence; we wish to show that $F$ is also a categorical equivalence. By virtue of Exercise 3.1.7.11, there exists a monomorphism of simplicial sets $\iota : \operatorname{\mathcal{C}}'' \hookrightarrow Q$, where $Q$ is a contractible Kan complex. Replacing $\overline{F}$ by the morphism $(\iota , \overline{F}): \operatorname{\mathcal{C}}'' \hookrightarrow Q \times \operatorname{\mathcal{C}}'$ (and $F$ by the morphism $(\iota ,F): \operatorname{\mathcal{D}}'' \hookrightarrow Q \times \operatorname{\mathcal{D}}'$), we can reduce to the case where $\overline{F}$ is a monomorphism of simplicial sets, so that $F$ is also a monomorphism of simplicial sets. To show that $F$ is a categorical equivalence, it will suffice to show that if $T_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}'_0$ is an isofibration of $\infty $-categories, then every lifting problem

7.17
\begin{equation} \begin{gathered}\label{equation:exponentiable-equivalent-condition2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}'' \ar [d]^{F} \ar [r] & \operatorname{\mathcal{E}}_0 \ar [d]^{T_0} \\ \operatorname{\mathcal{D}}' \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{E}}'_0 } \end{gathered} \end{equation}

admits a solution (Proposition 4.5.6.4). Invoking the universal property of direct images (Proposition 7.3.2.2), we can rewrite (7.17) as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'' \ar [d]^{ \overline{F} } \ar [r] & \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}_0 ) \ar [d] \\ \operatorname{\mathcal{C}}' \ar [r] \ar@ {-->}[ur] & \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}'_0 ). } \]

Condition $(4)$ guarantees that the right vertical map is an isofibration, so that the solution exists by virtue of our assumption that $\overline{F}$ is a categorical equivalence. $\square$

Corollary 7.3.2.19. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be an exponentiable morphism of simplicial sets. For every isofibration of simplicial sets $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$, the projection map $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ is also an isofibration of simplicial sets.

Proof. Applying Proposition 7.3.2.18 in the special case $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{D}}$. $\square$

We now formulate the main result of this section.

Proposition 7.3.2.20. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets. Then the projection map $\pi : \operatorname{Res}_{ \operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ is also a cartesian fibration of simplicial sets. Moreover, an edge $e$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ is $\pi $-cartesian if and only if it satisfies the following condition:

  • Form a commutative diagram

    \[ \xymatrix@R =50pt@C=50pt{ \Delta ^{1} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\ar [r]^-{ V_{e} } \ar [d] & \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\ar [r]^-{ U_{e} } \ar [d] & \Delta ^1 \ar [d]^{ \pi (e) } \\ \operatorname{\mathcal{E}}\ar [r]^-{V} & \operatorname{\mathcal{D}}\ar [r]^-{U} & \operatorname{\mathcal{C}}, } \]

    so that the edge $e$ can be identified with a morphism of simplicial sets $F_{e}: \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ such that $V_{e} \circ F_{e}$ is the identity. Then the morphism $F_{e}$ carries $U_{e}$-cocartesian morphisms of $\Delta ^{1} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ to $V_{e}$-cartesian morphisms of $\Delta ^{1} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.

We will carry out the proof of Proposition 7.3.2.20 in several steps.

Lemma 7.3.2.21. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets, and let $e$ be an edge of the simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ which satisfies condition $(\ast )$ of Proposition 7.3.2.20. Then $e$ is $\pi $-cartesian, where $\pi : \operatorname{Res}_{ \operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ denotes the projection map.

Proof. Let $n \geq 2$ and suppose we are given a lifting problem

7.18
\begin{equation} \begin{gathered}\label{equation:cartesian-in-direct-image} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{Res}_{ \operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \ar [d]^{\pi } \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}

where $\sigma _0$ carries the final edge $\operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subseteq \Lambda ^{n}_{n}$ to $e$; we wish to show that this lifting problem admits a solution. Replacing $U$ and $V$ with the projection maps $\Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^ n$ and $\Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, we can assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex and that $\overline{\sigma }: \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ is the identity map. Set $\operatorname{\mathcal{D}}_0 = \Lambda ^{n}_{n} \times _{\Delta ^ n} \operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}_0 = \Lambda ^{n}_{n} \times _{ \Delta ^{n} } \operatorname{\mathcal{E}}$. Invoking the universal property of the simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})$ (Proposition 7.3.2.2), we can rewrite (7.18) as a lifting problem

7.19
\begin{equation} \begin{gathered}\label{equation:cartesian-in-direct-image-3} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}_0 \ar [d] \ar [r]^-{F_0} & \operatorname{\mathcal{E}}\ar [d]^{V} \\ \operatorname{\mathcal{D}}\ar [r]^-{ \operatorname{id}_{\operatorname{\mathcal{D}}} } \ar@ {-->}[ur]^{F} & \operatorname{\mathcal{D}}. } \end{gathered} \end{equation}

Note that since the edge $e$ satisfies condition $(\ast )$, the diagram $F_0$ satisfies the following condition:

  • If $u$ is a $U$-cocartesian edge of $\operatorname{\mathcal{D}}$ lying over the final edge $\operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subseteq \operatorname{\mathcal{C}}$, then $F_0(u)$ is a $V$-cartesian edge of $\operatorname{\mathcal{E}}$.

For $0 \leq i \leq n$, let $\operatorname{\mathcal{D}}(i)$ denote the fiber $\{ i\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$. Applying Proposition 5.2.6.18 to the cocartesian fibration $U$, we conclude that there exist a scaffold $G: \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}' = M( \operatorname{\mathcal{D}}(0) \rightarrow \operatorname{\mathcal{D}}(1) \rightarrow \cdots \rightarrow \operatorname{\mathcal{D}}(n) )$ is the mapping simplex of some sequence of functors $\{ \operatorname{\mathcal{D}}(i-1) \rightarrow \operatorname{\mathcal{D}}(i) \} _{0 < i \leq n}$ (see Construction 5.2.6.3). Set $\operatorname{\mathcal{D}}'_0 = \Lambda ^{n}_{n} \times _{\Delta ^ n} \operatorname{\mathcal{D}}'$, so that $G$ restricts to a morphism $G_0: \operatorname{\mathcal{D}}'_0 \rightarrow \operatorname{\mathcal{D}}_0$. Note that we have a commutative diagram of $\infty $-categories

7.20
\begin{equation} \begin{gathered}\label{equation:cartesian-in-direct-image-2} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \ar [r]^-{\circ G} \ar [d] & \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}', \operatorname{\mathcal{E}}) \ar [d] \\ \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}_0, \operatorname{\mathcal{E}}) \ar [r]^-{\circ G_0} & \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}'_0, \operatorname{\mathcal{E}}). } \end{gathered} \end{equation}

Since $V$ is an isofibration (Proposition 5.1.4.8), the vertical maps in this diagram are isofibrations (Proposition 4.5.7.10). Since $G$ and $G_0$ are categorical equivalences of simplicial sets (Corollary 5.2.6.21), the horizontal maps are equivalences of $\infty $-categories. Applying Corollary 4.5.4.7, we deduce that the upper horizontal map in the diagram (7.20) restricts to an equivalence from each fiber of the left vertical map to the corresponding fiber of the right vertical map. Consequently, we can replace (7.19) with the lifting problem

7.21
\begin{equation} \begin{gathered}\label{equation:cartesian-in-direct-image-4} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}'_0 \ar [d] \ar [r]^-{F_0 \circ G_0} & \operatorname{\mathcal{E}}\ar [d]^{V} \\ \operatorname{\mathcal{D}}' \ar [r]^-{G} \ar@ {-->}[ur] & \operatorname{\mathcal{D}}. } \end{gathered} \end{equation}

Using Remark 5.2.6.9, we obtain a pushout square

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n} \times \operatorname{\mathcal{D}}(0) \ar [r]^-{H_0} \ar [d] & \operatorname{\mathcal{D}}'_0 \ar [d] \\ \Delta ^ n \times \operatorname{\mathcal{D}}(0) \ar [r]^-{H} & \operatorname{\mathcal{D}}'. } \]

Let us identify $F_0 \circ G_0 \circ H_0$ with a morphism of simplicial sets $\tau _0: \Lambda ^{n}_{n} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}(0), \operatorname{\mathcal{E}})$, and $G \circ H$ with an $n$-simplex $\overline{\tau }$ of $\operatorname{Fun}( \operatorname{\mathcal{D}}(0), \operatorname{\mathcal{D}})$, so that we can rewrite (7.21) again as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n} \ar [d] \ar [r]^-{ \tau _0 } & \operatorname{Fun}( \operatorname{\mathcal{D}}(0), \operatorname{\mathcal{E}}) \ar [d]^{V'} \\ \Delta ^{n} \ar [r]^-{\overline{\tau }} \ar@ {-->}[ur]^{\tau } & \operatorname{Fun}(\operatorname{\mathcal{D}}(0), \operatorname{\mathcal{D}}). } \]

To show that this lifting problem admits a solution, it will suffice to show that $\tau _0$ carries the final edge $\operatorname{N}_{\bullet }( \{ n-1 < n \} )$ of $\Lambda ^{n}_{n}$ to a $V'$-cocartesian edge of $\operatorname{Fun}( \operatorname{\mathcal{D}}(0), \operatorname{\mathcal{E}})$. This follows by combining $(\ast ')$ with the criterion of Theorem 5.2.1.1. $\square$

Lemma 7.3.2.22. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets, and let $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ denote the projection map. Suppose we are given a vertex $Y$ of the simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})$ having image $\overline{Y} = \pi (Y)$, and an edge $\overline{e}: \overline{X} \rightarrow \overline{Y}$ of the simplicial set $\operatorname{\mathcal{C}}$. Then we can write $\overline{e} = \pi (e)$ for some edge $e: X \rightarrow Y$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ which satisfies condition $(\ast )$ of Proposition 7.3.2.20.

Proof. As in the proof of Lemma 7.3.2.21, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$ and that $\overline{e}$ is the nondegenerate edge of $\operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{D}}(0)$ and $\operatorname{\mathcal{D}}(1)$ denote the fibers of $\operatorname{\mathcal{D}}$ over the vertices $\overline{X} = 0$ and $\overline{Y} = 1$, respectively, and let us identify $Y$ with a morphism of simplicial sets $\operatorname{\mathcal{D}}(1) \rightarrow \operatorname{\mathcal{E}}$. Applying Proposition 5.2.2.4, we can choose a functor $F: \operatorname{\mathcal{D}}(0) \rightarrow \operatorname{\mathcal{D}}(1)$ and a diagram

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{\mathcal{D}}(0) \ar [r]^-{H} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \Delta ^1 \ar@ {=}[r] & \operatorname{\mathcal{C}}} \]

which exhibits $F= H|_{ \{ 1\} \times \operatorname{\mathcal{D}}(0) }$ as given by covariant transport along $\overline{e}$. Applying Lemma 5.2.1.4 to the cartesian fibration $V$, we deduce that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{D}}(0) \ar [r]^-{Y \circ F} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{V} \\ \Delta ^1 \times \operatorname{\mathcal{D}}(0) \ar [r]^-{ H } \ar@ {-->}[ur]^{ \widetilde{H} } & \operatorname{\mathcal{D}}} \]

admits a solution with the property that, for every object $D$ of the $\infty $-category $\operatorname{\mathcal{D}}(0)$, the restriction $\widetilde{H}|_{ \Delta ^1 \times \{ D\} }$ is a $V$-cartesian morphism of $\operatorname{\mathcal{E}}$.

Let $\operatorname{\mathcal{D}}' = (\Delta ^1 \times \operatorname{\mathcal{D}}(0)) \coprod _{ (\{ 1\} \times \operatorname{\mathcal{D}}(0)) } \operatorname{\mathcal{D}}(1)$ denote the mapping cylinder of the functor $F$. Amalgamating $H$ with the inclusion map $\operatorname{\mathcal{D}}(1) \hookrightarrow \operatorname{\mathcal{D}}$, we obtain a morphism of simplicial sets $H': \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}$, which is a categorical equivalence by virtue of Corollary 5.2.5.2. Amalgamating $\widetilde{H}$ with $Y$, we obtain a diagram $\widetilde{H}': \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{E}}$ satisfying $V \circ \widetilde{H}' = H'$. We have a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \ar [r]^-{\circ G} \ar [d] & \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}', \operatorname{\mathcal{E}}) \ar [d] \\ \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}(1), \operatorname{\mathcal{E}}) \ar@ {=}[r] & \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}(1), \operatorname{\mathcal{E}}), } \]

where the horizontal maps are equivalences of $\infty $-categories. Since $V$ is an isofibration (Proposition 5.1.4.8), the vertical maps in this diagram are isofibrations (Proposition 4.5.7.10). Applying Corollary 4.5.4.7, we deduce that the upper horizontal map in the diagram (7.20) restricts to an equivalence of the fibers of the vertical maps over the object $Y \in \operatorname{Fun}_{ /\operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}(1), \operatorname{\mathcal{E}})$. It follows that there there exists a functor $E: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ such that $V \circ E = \operatorname{id}_{\operatorname{\mathcal{D}}}$, $E|_{\operatorname{\mathcal{D}}(1)} = Y$, and $E \circ G$ is isomorphic to $\widetilde{H}'$ as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}', \operatorname{\mathcal{E}})$. By construction, we can identify $E$ with an edge $e: X \rightarrow Y$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ satisfying $\pi (e) = \overline{e}$. To complete the proof, it will suffice to show that $e$ satisfies condition $(\ast )$ of Proposition 7.3.2.20. Let $u: D \rightarrow D'$ be a $U$-cocartesian edge of $\operatorname{\mathcal{D}}$ satisfying $\pi (u) = \overline{e}$; we wish to show that $E(u)$ is a $V$-cartesian edge of $\operatorname{\mathcal{E}}$. By virtue of Remark 5.1.3.8, we can assume without loss of generality that $u: D \rightarrow F(D)$ is the $U$-cocartesian morphism given by the restriction $\widetilde{F}|_{ \Delta ^1 \times \{ D\} }$. In this case, $E(u)$ is isomorphic (as an object of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{E}})$) to the $V$-cartesian morphism $\widetilde{H}|_{ \Delta ^1 \times \{ D\} }$, and is therefore also $V$-cartesian (Corollary 5.1.2.5). $\square$

Proof of Proposition 7.3.2.20. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets. Then $U$ is exponentiable (Example 7.3.2.16) and $V$ is an isofibration (Proposition 5.1.4.8), so the projection map $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ is also an isofibration (Corollary 7.3.2.19). In particular, $\pi $ is an inner fibration of simplicial sets (Remark 4.5.7.3). Let us say that an edge $e$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ is special if it satisfies condition $(\ast )$ of Proposition 7.3.2.20. Lemma 7.3.2.21 guarantees that every special edge of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ is $\pi $-cartesian. Moreover, if $Y$ is a vertex of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ and $\overline{e}: \overline{X} \rightarrow \pi (Y)$ is an edge of $\operatorname{\mathcal{C}}$, then Lemma 7.3.2.22 guarantees that there exists a special edge $e: X \rightarrow Y$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ satisfying $\pi (e) = \overline{e}$. It follows that $\pi $ is a cartesian fibration of simplicial sets.

To complete the proof of Proposition 7.3.2.20, we must show that every $\pi $-cartesian edge $e: X \rightarrow Y$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ is special. Without loss of generality we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$ and that $\pi (e)$ is the nondegenerate edge of $\operatorname{\mathcal{C}}$, so that we can identify $e$ with a functor $E: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ satisfying $V \circ E = \operatorname{id}_{\operatorname{\mathcal{D}}}$. Using Lemma 7.3.2.22, we can choose a special edge $e': X' \rightarrow Y$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ satisfying $\pi (e') = \pi (e)$, corresponding to another functor $E': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. Since $e'$ is also $\pi $-cartesian, it is isomorphic to $e$ as an object of the $\infty $-category $\operatorname{Fun}_{ / \Delta ^1 }( \Delta ^1, \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) )$, so that $E'$ is isomorphic to $E$ as an object of the $\infty $-category $\operatorname{Fun}_{ /\operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$. If $u$ is a $U$-cocartesian edge of $\operatorname{\mathcal{D}}$, then $E(u)$ is isomorphic to the $V$-cartesian morphism $E'(u)$ (as an object of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{E}})$), and is therefore also $V$-cartesian (Corollary 5.1.2.5). $\square$