Variant 8.3.3.5. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{S}}^{< \kappa }$ denote the $\infty $-category of $\kappa $-small spaces (Variant 5.5.4.13). Then an $\infty $-category $\operatorname{\mathcal{C}}$ admits a $\operatorname{Hom}$-functor
\[ \mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \]
if and only if it is locally $\kappa $-small. If this condition is satisfied, then $\mathscr {H}$ is uniquely determined up to isomorphism.