Kerodon

Example 8.3.4.14. In the situation of Definition 8.3.4.12, suppose that $\operatorname{\mathcal{D}}= \Delta ^0$. In this case, we can identify the profunctor $\mathscr {K}$ with a functor $K: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$, we can identify the functor $G$ with an object $X \in \operatorname{\mathcal{C}}$, and we can identify $\beta $ with a vertex of the Kan complex $K(X)$. Then $\beta $ exhibits the profunctor $\mathscr {K}$ as represented by the functor $G$ (in the sense of Definition 8.3.4.12) if and only if it exhibits the functor $K$ as represented by the object $X$ (in the sense of Variant 5.6.6.2.