Corollary 10.2.4.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits fiber products. The following conditions are equivalent:

- $(1)$
Every quotient morphism in $\operatorname{\mathcal{C}}$ is a universal quotient morphism.

- $(2)$
The collection of quotient morphisms in $\operatorname{\mathcal{C}}$ is closed under pullbacks. That is, for every pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y' \ar [r] & Y } \]where $f$ is a quotient morphism, $f'$ is also a quotient morphism.