Variant 10.3.4.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a $2$-simplex
If $f$ is a universal quotient morphism and $g$ is a quotient morphism, then $h$ is a quotient morphism.
Proof. Let $\operatorname{\mathcal{C}}^{0}_{/Z}$ and $\operatorname{\mathcal{C}}^{1}_{/Z}$ be the sieves on $X$ generated by $g$ and $h$, respectively. Our assumption that $g$ is a quotient morphism guarantees that the functor
is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, and we wish to show that the restriction $\overline{Q}|_{ (\operatorname{\mathcal{C}}^{1}_{/Z})^{\triangleright }}$ is also a colimit diagram. By virtue of Corollary 7.3.8.2, it will suffice to show that the restriction $Q = \overline{Q}|_{ \operatorname{\mathcal{C}}^{0}_{/Z} }$ is left Kan extended from the full subcategory $\operatorname{\mathcal{C}}^{1}_{/Z}$. Fix a morphism $u: Z' \rightarrow Z$ which belongs to the sieve $\operatorname{\mathcal{C}}^{0}_{/Z}$; we wish to show that $Q$ is left Kan extended from $\operatorname{\mathcal{C}}^{1}_{/Z}$ at $u$. In fact, we will prove a slightly stronger assertion: the pullback $u^{\ast }( \operatorname{\mathcal{C}}^{1}_{/Z} )$ is a dense sieve on $Z'$. Using Proposition 10.3.1.33, we are reduced to proving this in the special case where $u$ is the morphism $g: Y \rightarrow Z$. In this case, the sieve $u^{\ast }( \operatorname{\mathcal{C}}^{1}_{/Z} )$ contains the quotient morphism $f$, and is therefore dense by virtue of Remark 10.3.4.4. $\square$