Corollary 10.3.4.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the collection of universal quotient morphisms of $\operatorname{\mathcal{C}}$ is closed under retracts (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $f: X \twoheadrightarrow Y$ be a universal quotient morphism in $\operatorname{\mathcal{C}}$ and let $f': X' \rightarrow Y'$ be a retract of $f$, so that we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \ar [r]^-{r_{X}} & X' \ar [d]^{f'} \\ Y' \ar [r] & Y \ar [r]^-{ r_{Y} } & Y' } \]
where the vertical compositions are homotopic to the identity. We wish to show that $f'$ is also a universal quotient morphism. By virtue of Remark 10.3.4.5, it will suffice to show that the composition $(f' \circ r_{X}): X \rightarrow Y'$ is a universal quotient morphism. Using the commutativity of the diagram, we can write $f' \circ r_{X}$ as a composition of $r_{Y}$ with $f$. Since $f$ is a universal quotient morphism by assumption and $r_{Y}$ is a universal quotient morphism by virtue of Example 10.3.4.3, the desired result follows from Proposition 10.3.4.12. $\square$