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Lemma 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 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

\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, we can rewrite (5.30) as a lifting problem

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

Using Corollary, we can choose a diagram of $\infty $-categories $\mathscr {F}: [n] \rightarrow \operatorname{QCat}$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{D}}$. Set $\operatorname{\mathcal{D}}' = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ and $\operatorname{\mathcal{D}}'_0 = \Lambda ^{n}_{n} \times _{ \Delta ^ n } \operatorname{\mathcal{D}}'$, so that $\lambda $ restricts to a map $\lambda _0: \operatorname{\mathcal{D}}'_0 \rightarrow \operatorname{\mathcal{D}}_0$. We then have a commutative diagram of $\infty $-categories

\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 \lambda } \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 \lambda } & \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}'_0, \operatorname{\mathcal{E}}). } \end{gathered} \end{equation}

Since $V$ is an isofibration (Proposition, the vertical maps in this diagram are isofibrations (Proposition Since $\lambda $ and $\lambda _0$ are categorical equivalences of simplicial sets (Lemma, the horizontal maps are equivalences of $\infty $-categories. Applying Corollary, we deduce that the upper horizontal map in the diagram (5.32) 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 (5.31) with the lifting problem

\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, we obtain a pushout square

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

Let us identify $F_0 \circ G_0 \circ H_0$ with a morphism of simplicial sets $\tau _0: \Lambda ^{n}_{n} \rightarrow \operatorname{Fun}( \mathscr {F}(0), \operatorname{\mathcal{E}})$, and $G \circ H$ with an $n$-simplex $\overline{\tau }$ of $\operatorname{Fun}( \mathscr {F}(0), \operatorname{\mathcal{D}})$, so that we can rewrite (5.33) again as a lifting problem

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

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 $\operatorname{Fun}( \mathscr {F}(0), \operatorname{\mathcal{E}})$. Since $\lambda $ is a scaffold, this follows by combining $(\ast ')$ with the criterion of Theorem $\square$