Kerodon

Warning 8.3.2.10. The terminology of Definition 8.3.2.9 is potentially confusing. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories, let $\operatorname{\mathcal{C}}$ denote the product $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$, and let $\mathscr {K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a morphism of simplicial sets. In general, there is no relationship between the corepresentability of $\mathscr {K}$ as a $\operatorname{\mathcal{S}}$-valued functor on $\operatorname{\mathcal{C}}$ (in the sense of Definition 5.6.6.1) and the corepresentability of $\mathscr {K}$ as a profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$ (in the sense of Definition 8.3.2.9). However, these notions of corepresentability coincide when $\operatorname{\mathcal{C}}_{-}$ is a contractible Kan complex (see Example 8.3.2.13).