Kerodon

Remark 8.3.4.7 (Uniqueness). Let $\kappa $ be an uncountable cardinal and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ be a profunctor of $\infty $-categories. If $\operatorname{\mathcal{C}}$ is locally $\kappa $-small small, then Proposition 8.3.4.1 guarantees that $\mathscr {K}$ is representable (in the sense of Definition 8.3.2.9 if and only if it is representable by $G$, for some functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Moreover, if this condition is satisfied, then the functor $G$ is determined uniquely to up isomorphism. Similarly, if $\operatorname{\mathcal{D}}$ is locally $\kappa $-small small, then $\mathscr {K}$ is corepresentable if and only if it is corepresentable by some functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. In this case, the functor $F$ is also uniquely determined up to isomorphism.