Kerodon

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Example 5.6.0.8. Let $\operatorname{\mathcal{C}}$ be a small $\infty$-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. It follows from Corollary 5.6.0.7 that there is an essentially unique functor $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ for $\int _{\operatorname{\mathcal{C}}} h^{X}$ is equivalent to $\operatorname{\mathcal{C}}_{X/}$ as left fibrations over $\operatorname{\mathcal{C}}$. We will refer to $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as the functor corepresented by $X$. For every object $Y \in \operatorname{\mathcal{C}}$, we have isomorphisms

$h^{X}(Y) \simeq \{ Y\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} h^{X} \simeq \{ Y\} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/} = \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, depending functorially on $Y$. In §5.6.6, we will show that this property characterizes the functor $h^{X}$ up to isomorphism (Theorem 5.6.6.14).