Kerodon

Remark 11.5.0.72. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let us regard the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as enriched over the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Construction 4.6.9.13). Every functor of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ induces an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{S}}} = \mathrm{h} \mathit{\operatorname{Kan}}$. Moreover, if $X$ is an object of $\operatorname{\mathcal{C}}$, then every vertex $x \in \mathscr {F}(X)$ determines a natural transformation of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, \bullet ) \rightarrow \mathrm{h} \mathit{\mathscr {F}}(\bullet )$, which is an isomorphism if and only if $x$ exhibits $\mathscr {F}$ as corepresented by $X$. Consequently, $\mathscr {F}$ is corepresentable by $X$ if and only if $\mathrm{h} \mathit{\mathscr {F}}$ is isomorphic to $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, \bullet )$ as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor.