Remark 5.6.6.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then every object $X \in \operatorname{\mathcal{C}}$ determines an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor
Theorem 5.6.6.13 asserts $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, \bullet )$ can be promoted, in an essentially unique way, to a functor of $\infty $-categories $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ (see Remark 5.6.6.12). Beware that this is a special feature of corepresentable functors. In general, an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\mathscr {F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ cannot be promoted to a functor of $\infty $-categories. Moreover, when such a promotion exists, it need not be unique.