Kerodon

Warning 5.6.6.9. The converse of Remark 5.6.6.8 is false in general. For example, let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}\subset \operatorname{\mathcal{S}}$ be the functor given on objects by the formula $\mathscr {F}(Y) = \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) )$. Then $\pi _0(\mathscr {F} )$ is corepresentable by the object $X$ (when regarded as a functor from $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the category of sets), but the functor $\mathscr {F}$ is usually not corepresentable.