Proposition 8.1.9.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories, let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which are pushout-compatible, and assume that morphisms of $R$ admit $U$-cocartesian lifts. Let $\widetilde{L}$ denote the collection of all morphisms $f$ of $\operatorname{\mathcal{E}}$ such that $U(f) \in L$, and let $\widetilde{R}$ denote the collection of all $U$-cocartesian morphisms $f$ of $\operatorname{\mathcal{E}}$ such that $U(f) \in R$. Then the collections $\widetilde{L}$ and $\widetilde{R}$ are also pushout-compatible.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $f: X \rightarrow X_1$ be a morphism of $\operatorname{\mathcal{E}}$ which belongs to $\widetilde{L}$, and let $g': X \rightarrow X_0$ be a morphism of $\operatorname{\mathcal{E}}$ which belongs to $\widetilde{R}$. We wish to show that there exists a pushout diagram
8.25
\begin{equation} \begin{gathered}\label{equation:dual-Beck-Chevalley-nonsense} \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{g'} \ar [d]^{f} & X_0 \ar [d]^{f'} \\ X_1 \ar [r]^-{g} & X_{01} } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{E}}$, where $f'$ belongs to $\widetilde{L}$ and $g$ belongs to $\widetilde{R}$. Since $L$ and $R$ are pushout compatible, there exists a pushout diagram
8.26
\begin{equation} \begin{gathered}\label{equation:dual-Beck-Chevalley-nonsense2} \xymatrix@R =50pt@C=50pt{ U(X) \ar [r]^-{ U(g') } \ar [d]^{ U(f) } & U(X_0) \ar [d]^{ \overline{f}' } \ar [d] \\ U(X_1) \ar [r]^-{ \overline{g}} & \overline{X}_{01} } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{C}}$, where $\overline{f}'$ belongs to $L$ and $\overline{g}$ belongs to $R$. Our assumption on $U$ guarantees that $\overline{g}$ can be lifted to a $U$-cocartesian morphism $g: X_0 \rightarrow X_{01}$ of $\operatorname{\mathcal{E}}$. Since $U$ is an inner fibration, the lower left half of (8.26) can be lifted to a $2$-simplex $\sigma $ of $\operatorname{\mathcal{E}}$ which we display as a diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [dr]^{ h } & \\ X_1 \ar [r]^-{g} & X_{01}. } \]
Since $g'$ is $U$-cocartesian, we can then lift the upper right half of (8.26) to a $2$-simplex $\tau $ of $\operatorname{\mathcal{E}}$ which we display as a diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{g'} \ar [dr]^{ h } & X_0 \ar [d]^{f'} \\ & X_{01}. } \]
Amalgamating $\sigma $ and $\tau $, we obtain a diagram of the form (8.25), where $f' \in \widetilde{L}$ and $g \in \widetilde{R}$. We will complete the proof by showing that this diagram is a pushout square in the $\infty $-category $\operatorname{\mathcal{E}}$. Since (8.26) is a pushout square in $\operatorname{\mathcal{C}}$, it will suffice to show that 8.25 is a $U$-pushout square (Corollary 7.1.6.18). This is a special case of Proposition 7.6.2.26, since the horizontal morphisms appearing in the diagram are $U$-cocartesian. $\square$