Remark 8.6.6.10 (Existence and Uniqueness). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $\kappa $ be an uncountable cardinal. Then $U$ admits a relative $\operatorname{Hom}$-functor
\[ \mathscr {H}: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa } \]
if and only if the inner fibration $U$ is locally $\kappa $-small. Moreover, if this condition is satisfied, then $\mathscr {H}$ is unique up to isomorphism (Corollary 5.6.0.6).