$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 9.2.1.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given a pushout diagram
9.4
\begin{equation} \begin{gathered}\label{equation:localizing-morphisms-closed-under-pushout} \xymatrix@C =50pt@R=50pt{ X \ar [r]^-{w} \ar [d] & Y \ar [d] \\ X' \ar [r]^-{w'} & Y' } \end{gathered} \end{equation}
in $\operatorname{\mathcal{C}}$. If $W$ is a localizing collection of morphisms of $\operatorname{\mathcal{C}}$ which contains $w$, then it also contains $w'$.
Proof.
Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory of $W$-local objects and let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a left adjoint to the inclusion. The assumption $w \in W$ guarantees that $L(w)$ is an isomorphism in $\operatorname{\mathcal{C}}'$ (Proposition 9.2.1.18). Since $L$ carries the (9.4) to a pushout diagram in the $\infty $-category $\operatorname{\mathcal{C}}'$ (Corollary 7.1.4.22), it follows that $L(w')$ is also an isomorphism (Corollary 7.6.2.27). Applying Proposition 9.2.1.18 again, we conclude that $w'$ belongs to $W$.
$\square$