Remark 8.6.6.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories and let
\[ \mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa } \]
be a relative $\operatorname{Hom}$-functor for $U$. Then, for every vertex $C \in \operatorname{\mathcal{C}}$, the restriction of $\mathscr {H}$ to
\[ \operatorname{\mathcal{E}}_{C}^{\operatorname{op}} \times \operatorname{\mathcal{E}}_{C} \simeq \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ \operatorname{id}_ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}} \]
is $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, in the sense of Definition 8.3.3.1.