# Kerodon

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Definition 5.6.6.1 (Corepresentable Functors). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing an object $X$, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor, and let $x$ be a vertex of the Kan complex $\mathscr {F}(X)$. We will say that $x$ exhibits $\mathscr {F}$ as corepresented by $X$ if, for every object $Y \in \operatorname{\mathcal{C}}$, the composite map

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & \xrightarrow {\mathscr {F}} & \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \mathscr {F}(X), \mathscr {F}(Y) ) \\ & \simeq & \operatorname{Fun}( \mathscr {F}(X), \mathscr {F}(Y) ) \\ & \xrightarrow {\operatorname{ev}_ x} & \mathscr {F}(Y) \end{eqnarray*}

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$; here the second map is the inverse of the homotopy equivalence $\operatorname{Fun}( \mathscr {F}(X), \mathscr {F}(Y) ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( X, Y)$ supplied by Remark 5.5.1.5.

We say that the functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by $X$ if there exists a vertex $x \in \mathscr {F}(X)$ which exhibits $\mathscr {F}$ as corepresented by $X$. We say that the functor $\mathscr {F}$ is corepresentable if it is corepresentable by $X$, for some object $X \in \operatorname{\mathcal{C}}$.