Lemma 10.3.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $q: X \twoheadrightarrow Y$ be a quotient morphism in $\operatorname{\mathcal{C}}$. Then $q$ is left orthogonal to every monomorphism $i: C \hookrightarrow D$ of $\operatorname{\mathcal{C}}$.
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Proof. Let $U: \operatorname{\mathcal{C}}_{/D} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, so that the monomorphism $i$ can be identified with a subterminal object $\widetilde{C} \in \operatorname{\mathcal{C}}_{/D}$ satisfying $U( \widetilde{C} ) = C$ (Remark 9.3.4.16). By virtue of Corollary 9.2.7.13, it will suffice to show that the object $\widetilde{C}$ is $\widetilde{q}$-local for every morphism $\widetilde{q}$ of $\operatorname{\mathcal{C}}_{/D}$ satisfying $U( \widetilde{q} ) = q$. This follows from Lemma 10.3.3.9, since $\widetilde{q}$ is a quotient morphism in the $\infty $-category $\operatorname{\mathcal{C}}_{/D}$ (Proposition 10.3.2.14). $\square$