Remark 10.3.1.32. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $Y$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be a sieve on $X$. Let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, which we regard as an object of $\operatorname{\mathcal{C}}_{/Y}$, and let $\operatorname{\mathcal{C}}^{0}_{/X} = f^{\ast }( \operatorname{\mathcal{C}}^{0}_{/Y})$ be the pullback sieve (Notation 10.3.1.24). Then the forgetful functor $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}_{/Y}$ at $f$ if and only if the following condition is satisfied:
- $(\ast _{f})$
The composite map
\[ ( \operatorname{\mathcal{C}}_{/X}^{0} )^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright }_{/X} \rightarrow \operatorname{\mathcal{C}} \]is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.
In particular, the sieve $\operatorname{\mathcal{C}}^{0}_{/Y}$ is dense if and only if it satisfies condition $(\ast _{f})$ for every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$.