Kerodon

Example 8.6.6.11. Let $\operatorname{\mathcal{E}}$ be a locally $\kappa $-small $\infty $-category and let $\mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{E}}$. Suppose we are given a functor $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(J)$ is the nerve of a partially ordered set $J$. Then $U$ is automatically an inner fibration (Proposition 4.1.1.10). Moreover, the iterated fiber product $\operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ can be identified with the full subcategory of $\operatorname{\mathcal{E}}^{\operatorname{op}} \times \operatorname{\mathcal{E}}$ spanned by those objects $(X,Y)$ satisfying $U(X) \leq U(Y)$ (see Example 8.1.0.5). In this case, the restriction of $\mathscr {H}$ to this full subcategory is a relative $\operatorname{Hom}$-functor for the inner fibration $U$, in the sense of Definition 8.6.6.8.