Kerodon

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Remark 6.2.4.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $U: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ be a functor of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched categories. Then the construction $C \mapsto \pi _0( U(C) )$ determines a functor from $\mathrm{h} \mathit{\operatorname{Kan}}$ to the category of sets, which we will denote by $\pi _0(U)$. If $X \in \operatorname{\mathcal{C}}$ represents the functor $U$ (in the sense of Definition 6.2.4.1), then it also represents the set-valued functor $\pi _0(U)$: that is, there are bijections $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, X) \simeq \pi _0( U(C) )$ which depend functorially on $C$. In particular, if the functor $U$ is representable (in the sense of Definition 6.2.4.1), then the functor $\pi _0(U)$ is also representable. Beware that the converse is false in general.