Definition 5.6.6.10. Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.5.10) and let $\operatorname{\mathcal{C}}$ be an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category containing an object $X$. We will say that an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ is corepresentable by $X$ if there exists a vertex $x \in \mathscr {F}(X)$ such that, for every object $Y \in \operatorname{\mathcal{C}}$, the induced map
is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. In this case, we also say that $x$ exhibits $\mathscr {F}$ as corepresented by the object $X$. We say that the functor $\mathscr {F}$ is corepresentable if it is corepresentable by $X$ for some object $X \in \operatorname{\mathcal{C}}$.
We say that an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ is representable by $X$ if it is corepresentable by the object $X^{\operatorname{op}} \in \operatorname{\mathcal{C}}^{\operatorname{op}}$, and that $\mathscr {F}$ is representable if it is representable by some object of $\operatorname{\mathcal{C}}$.