Proposition 8.6.4.18. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets. Let $\kappa $ be an uncountable cardinal such that $U$ is locally $\kappa $-small. Fix a morphism $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, which we identify with a morphism $F: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Then:
- $(1)$
The morphism $\mathscr {K}$ is a weak $\operatorname{\mathcal{C}}$-family of corepresentable profunctors if and only if $F$ factors through the simplicial subset $\operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{<\kappa }) \subseteq \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$.
- $(2)$
The morphism $\mathscr {K}$ is a $\operatorname{\mathcal{C}}$-family of corepresentable profunctors if and only if $F$ factors through $\operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ and carries $U^{\vee }$-cocartesian edges of $\operatorname{\mathcal{E}}^{\vee }$ to $\pi ^{\operatorname{corep}}$-cocartesian edges of $\operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$. Here $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{\mathcal{C}}$ denotes the cocartesian fibration of Proposition 8.6.4.12.
- $(3)$
The morphism $\mathscr {K}$ exhibits $U^{\vee }$ as a cocartesian dual of $U$ if and only if $F: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$.