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Remark 5.6.6.11. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor of $\infty $-categories, and let $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{S}}} = \mathrm{h} \mathit{\operatorname{Kan}}$ be the induced functor of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched homotopy categories (see Construction 4.6.8.13). Then:

  • A vertex $x \in \mathscr {F}(X)$ exhibits $\mathscr {F}$ as corepresented by an object $X \in \operatorname{\mathcal{C}}$ (in the sense of Definition 5.6.6.1) if and only if it exhibits $\mathrm{h} \mathit{\mathscr {F}}$ as corepresented by the object $X \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (in the sense of Definition 5.6.6.10).

  • The functor $\mathscr {F}$ is corepresentable by an object $X \in \operatorname{\mathcal{C}}$ if and only if $\mathrm{h} \mathit{\mathscr {F}}$ is corepresentable by $X \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

  • The functor $\mathscr {F}$ is corepresentable if and only if $\mathrm{h} \mathit{\mathscr {F}}$ is corepresentable as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor.