Definition 8.6.5.16. 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 and let $\kappa $ be an uncountable cardinal. We will say that a morphism $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ is a weak $\operatorname{\mathcal{C}}$-family of corepresentable profunctors if, for every vertex $C \in \operatorname{\mathcal{C}}$, the induced map
is a corepresentable profunctor (Definition 8.3.2.9). We say that $\mathscr {K}$ is a $\operatorname{\mathcal{C}}$-family of corepresentable profunctors if it is a weak $\operatorname{\mathcal{C}}$-family of corepresentable profunctors and satisfies the following additional condition:
- $(\ast )$
Let $f: X \rightarrow Y$ be a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ and let $f^{\vee }: X^{\vee } \rightarrow Y^{\vee }$ be a $U^{\vee }$-cocartesian edge of $\operatorname{\mathcal{E}}^{\vee }$ having the same image $u: C \rightarrow D$ in $\operatorname{\mathcal{C}}$. Then the map of Kan complexes
\[ \mathscr {K}( f^{\vee }, f ): \mathscr {K}_{C}( X^{\vee }, X) \rightarrow \mathscr {K}_{D}( Y^{\vee }, Y ) \]carries couniversal vertices of $\mathscr {K}_{C}( X^{\vee }, X )$ to couniversal vertices of $\mathscr {K}_{D}( Y^{\vee }, Y)$.