Lemma 10.3.4.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a dense full subcategory, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Suppose that, for every object $C \in \operatorname{\mathcal{C}}'$, postcomposition with $[f]$ induces a surjection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, X ) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, Y )$. Then $f$ is a universal quotient morphism.
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Proof. Our assumption that $\operatorname{\mathcal{C}}'$ is dense guarantees that the identity functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is left Kan extended from $\operatorname{\mathcal{C}}'$. Let $U: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map and let $\operatorname{\mathcal{C}}'_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ denote the inverse image of $\operatorname{\mathcal{C}}'$. Since $U$ is a right fibration (Proposition 4.3.6.1), the functor $U$ is left Kan extended from $\operatorname{\mathcal{C}}'_{/Y}$. Let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ denote the sieve generated by $f$. Our hypothesis guarantees that $\operatorname{\mathcal{C}}^{0}_{/Y}$ contains $\operatorname{\mathcal{C}}'_{/Y}$. Applying Corollary 7.3.8.8, we conclude that $U$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}_{/Y}$: that is, $f$ is a universal quotient morphism. $\square$