Kerodon

Variant 5.6.6.2 (Representable Functors). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and write $X^{\operatorname{op}}$ for the corresponding object of the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Given a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$, we say that a vertex $x \in \mathscr {F}(X^{\operatorname{op}} )$ exhibits $\mathscr {F}$ as represented by $X$ if it exhibits $\mathscr {F}$ as corepresented by the object $X^{\operatorname{op}}$, in the sense of Definition 5.6.6.1. We say that a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable by $X$ if it is corepresentable by $X^{\operatorname{op}}$, and that $\mathscr {F}$ is representable if it is representable by $X$ for some object $X \in \operatorname{\mathcal{C}}$.