Corollary 10.3.3.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which has images, and let
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]
be a $2$-simplex of $\operatorname{\mathcal{C}}$, where $f$ is a quotient morphism. Then $g$ is a quotient morphism if and only if $h$ is a quotient morphism. In particular, the collection of quotient morphisms is closed under composition.