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Lemma 5.3.6.8. 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 5.3.6.6.

Proof. As in the proof of Lemma 5.3.6.7, 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 5.2.2.8, 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 5.2.4.2. 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 4.5.5.14). Applying Corollary 4.5.2.26, we deduce that the upper horizontal map in the diagram (5.32) 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 5.3.6.6. 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). $\square$