Lemma 9.4.8.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty $-categories, let $\kappa $ be an uncountable regular cardinal for which $U$ is essentially $\kappa $-small, and let $H: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a relative $\operatorname{Hom}$-functor for $U$. Then $H$ is classified by a functor $h: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger }$.
Proof. Fix an object $X \in \operatorname{\mathcal{E}}$ having image $C = U(X)$ and an edge $e$ of the $\infty $-category $\operatorname{\mathcal{C}}_{C/}$, which we identify with a $2$-simplex $\sigma :$
of $\operatorname{\mathcal{C}}$, and let $H_{e}: \operatorname{N}_{\bullet }( \{ C' < C'' \} ) \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be as in condition $(\ast )$ of Remark 9.4.8.5; we wish to show that the functor $F_{e}$ is left Kan extended from the subcategory $\{ C' \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^2$ and that $\sigma $ is the identity morphism. In this case, we can identify $F_{e}$ with the composition
where $\mathscr {F}$ is corepresented by the object $X \in \operatorname{\mathcal{E}}$. Since $U$ is flat, the desired result is a reformulation of Lemma 9.4.7.2. $\square$