$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 10.3.4.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pullback diagram
10.26
\begin{equation} \begin{gathered}\label{equation:pullback-of-quotient} \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y' \ar [r]^-{u} & Y. } \end{gathered} \end{equation}
If $f$ is a universal quotient morphism, then $f'$ is also a universal quotient morphism.
Proof.
Let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be the sieve generated by $f$. Our assumption that $f$ is a universal quotient morphism guarantees that $\operatorname{\mathcal{C}}^{0}_{/Y}$ is a dense sieve on $Y$. Applying Proposition 10.3.1.33, we deduce that the pullback $u^{\ast } \operatorname{\mathcal{C}}^{0}_{/Y}$ is a dense sieve on $Y'$. Since (10.26) is a pullback square, the sieve $u^{\ast } \operatorname{\mathcal{C}}^{0}_{/Y'}$ is generated by $f'$, so that $f'$ is also a universal quotient morphism.
$\square$