Kerodon

Remark 8.4.1.21. Let $\kappa $ be an uncountable regular cardinal and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Assume that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small and that $\operatorname{\mathcal{D}}$ is a locally $\kappa $-small $\infty $-category. Then, for each object $X \in \operatorname{\mathcal{D}}$, the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X}$ is also essentially $\kappa $-small (Corollary 5.6.7.7). Let $\lambda $ be an infinite cardinal satisfying $\mathrm{ecf}(\lambda ) \geq \kappa $ (see Definition 4.7.3.16). Then $F$ is dense if and only if, for every representable functor $h_{Y}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$, the identity transformation $\operatorname{id}: h_{Y} \circ F^{\operatorname{op}} \rightarrow h_{Y} \circ F^{\operatorname{op}}$ exhibits the functor $h_{Y}$ as a right Kan extension of $h_{Y} \circ F^{\operatorname{op}}$ along $F^{\operatorname{op}}$. This follows by combining Remark 8.4.1.20 with Proposition 7.4.5.16 (together with Remark 7.4.5.18).