Example 9.4.2.5 (05LL)

Example 9.4.2.5. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets which is essentially $\kappa $-small, and let $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be as in Construction 8.6.5.6. It follows from Proposition 8.6.5.8 that the evaluation functor

\[ \operatorname{ev}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad ((C, \mathscr {F}_{C}), X) \mapsto \mathscr {F}_{C}(X) \]

exhibits $U$ as a cocartesian dual of the projection map $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{\mathcal{C}}$. Applying Lemma 9.4.2.4 (with the roles of $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}^{\vee }$ switched) and Lemma 9.4.2.3, we conclude that the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \ar [rr] \ar [dr] & & \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \ar [dl] \\ & \operatorname{\mathcal{C}}& } \]

exhibits $\operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ as a fiberwise $\kappa $-cocompletion of $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$.