Lemma 9.4.2.4. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be essentially $\kappa $-small cocartesian fibrations of simplicial sets, and let $\operatorname{Fun}( \operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be the relative exponential of Construction 4.5.9.1. Suppose we are given a morphism
\[ \mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }, \]
which exhibits $U^{\vee }$ as a cocartesian dual of $U$ (see Variant 8.6.4.13). Then $\mathscr {K}$ is classified by a commutative diagram with a diagram
9.37
\begin{equation} \begin{gathered}\label{equation:describe-relative-cocompletion} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \operatorname{Fun}( \operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}. & } \end{gathered} \end{equation}
which satisfies the conditions of Theorem 9.4.2.1.