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Definition 6.2.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote the homotopy category of $\operatorname{\mathcal{C}}$, which we regard as enriched over the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$ (Construction 4.6.7.13). Let $U: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ be an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor. We will say that $U$ is representable if there exists an object $X \in \operatorname{\mathcal{C}}$ and a vertex $u \in U(X)$ with the following property: for every object $C \in \operatorname{\mathcal{C}}$, the composite map

\[ \{ u \} \times \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, X ) \hookrightarrow \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, X ) \times U(X) \rightarrow U(C) \]

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. In this case, we say that the object $X \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ represents the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $U$.

We say that an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $V: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ is corepresentable if there exists an object $Y \in \operatorname{\mathcal{C}}$ and a vertex $v \in V(Y)$ with the property that, for every object $C \in \operatorname{\mathcal{C}}$, the composite map

\[ \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, C ) \times \{ v \} \hookrightarrow \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, C ) \times V(Y) \rightarrow V(C) \]

is an isomorphism in $\mathrm{h} \mathit{\operatorname{Kan}}$. In this case, we will say that the object $Y \in \operatorname{\mathcal{C}}$ corepresents the functor $V$.