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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.

Proof. It follows from Proposition 8.6.5.12 that $\widehat{U}$ is both a cartesian and cocartesian fibration, so that condition $(b)$ of Theorem 9.4.2.1 is satisfied and condition $(c)$ is automatic (Corollary 7.1.4.22). Condition $(d)$ follows from Proposition 8.6.5.18, and condition $(a)$ from Theorem 8.4.3.3. $\square$