Kerodon

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Remark 5.7.6.8. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor of $\infty $-categories. Then the construction $Y \mapsto \pi _0( \mathscr {F}(Y) )$ determines a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the category of sets, which we will denote by $\pi _0( \mathscr {F} )$. Suppose that $X$ is an object of $\operatorname{\mathcal{C}}$ and $x \in \mathscr {F}(X)$ exhibits $\mathscr {F}$ as corepresented by $X$. Then, for every object $Y \in \operatorname{\mathcal{C}}$, evaluation on the connected component $[x] \in \pi _0( \mathscr {F}(X) )$ induces a bijection

\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) = \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) ) \rightarrow \pi _0( \mathscr {F}(Y) ). \]

It follows that the functor $\pi _0(\mathscr {F}): \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ is corepresentable by $X$, in the sense of classical category theory.