Variant 9.3.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pushout square
9.8
\begin{equation} \begin{gathered}\label{proposition:w-local-pushout} \xymatrix@C =50pt@R=50pt{ X \ar [d]^{w} \ar [r] & X' \ar [d]^{w'} \\ Y \ar [r] & Y'. } \end{gathered} \end{equation}
If an object $C \in \operatorname{\mathcal{C}}$ is $w$-local, then it is also $w'$-local. This follows from Corollary 7.6.2.27, together with the observation that the representable functor $h_{C}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ carries pushout diagrams in $\operatorname{\mathcal{C}}$ to pullback diagrams in $\operatorname{\mathcal{S}}$ (Corollary 7.4.1.19).