Proposition 10.3.1.29. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pairwise products and let $X \in \operatorname{\mathcal{C}}$. Then a sieve $\operatorname{\mathcal{C}}^{0}_{/X} \subseteq \operatorname{\mathcal{C}}_{/X}$ is dense (in the sense of Definition 10.3.1.26) if and only if it is dense when regarded as a subcategory of $\operatorname{\mathcal{C}}_{/X}$ (in the sense of Definition 8.4.1.5).
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Proof. Assume that $\operatorname{\mathcal{C}}^{0}_{/X}$ is a dense subcategory of $\operatorname{\mathcal{C}}_{/X}$; we will show that it is dense when regarded as a sieve (for the reverse implication, see Warning 10.3.1.27). By assumption, the identity functor $\operatorname{id}: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}_{/X}$ is left Kan extended from $\operatorname{\mathcal{C}}^0_{/X}$. We wish to show that the forgetful functor $U: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ is also left Kan extended from $\operatorname{\mathcal{C}}^{0}_{/X}$. To prove this, it suffices to show that the functor $U$ preserves colimits. This is a special case of Corollary 7.1.4.22, since the functor $U$ admits a right adjoint (given on objects by the construction $Y \mapsto X \times Y$; see Proposition 7.6.1.14). $\square$