Corollary 10.2.2.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_0$ and $f_1$ be morphisms of $\operatorname{\mathcal{C}}$ which are isomorphic (when viewed as objects of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$). Then $f_0$ is a quotient morphism if and only if $f_1$ is a quotient morphism.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms. By virtue of Corollary 4.4.5.10, the evaluation functors $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ are equivalences of $\infty $-categories. Our assumption that $f_0$ is isomorphic to $f_1$ guarantees that there exists a morphism $\widetilde{f}$ of $\operatorname{Isom}(\operatorname{\mathcal{C}})$ satisfying $\operatorname{ev}_0( \widetilde{f} ) = f_0$ and $\operatorname{ev}_1( \widetilde{f} ) = f_1$. Using Proposition 10.2.2.11, we see that the condition that $f_0$ is a quotient morphism in $\operatorname{\mathcal{C}}$ is equivalent to the condition that $\widetilde{f}$ is a quotient morphism in $\operatorname{Isom}(\operatorname{\mathcal{C}})$, which is also equivalent to the condition that $f_1$ is a quotient morphism in $\operatorname{\mathcal{C}}$. $\square$