Proposition 10.3.4.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a $2$-simplex
If $f$ and $g$ are universal quotient morphisms, then $h$ is also a universal quotient morphism.
Proof. Let $\operatorname{\mathcal{C}}^{0}_{/Z}$ and $\operatorname{\mathcal{C}}^{1}_{/Z}$ be the sieves generated by $g$ and $h$, respectively. By assumption, the sieve $\operatorname{\mathcal{C}}^{0}_{/Z}$ is dense, and we wish to show that $\operatorname{\mathcal{C}}^{1}_{/Z}$ is also dense. By virtue of Proposition 10.3.1.34, it will suffice to show that for every morphism $u: Z' \rightarrow Z$ which belongs to $\operatorname{\mathcal{C}}^{0}_{/Z}$, the pullback $u^{\ast } \operatorname{\mathcal{C}}^{1}_{/Z'}$ is a dense sieve on $Z'$. Using Proposition 10.3.1.33, we can reduce to the special case where $u$ is the morphism $g: Y \twoheadrightarrow Z$. In this case, the pullback sieve $u^{\ast }( \operatorname{\mathcal{C}}^{1}_{/Y}) \subseteq \operatorname{\mathcal{C}}_{/Y}$ contains the universal quotient morphism $f: X \twoheadrightarrow Y$, and is therefore dense (Remark 10.3.4.4). $\square$