Kerodon

Example 7.6.6.33. Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Choose an uncountable cardinal $\lambda $ for which $U$ is essentially $\lambda $-small, so that $U$ admits a covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$. Then the cocartesian fibration $U$ is $\mathbb {K}$-cocomplete if and only if $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ factors through the subcategory $\operatorname{\mathcal{QC}}_{< \lambda }^{\mathbb {K}-\mathrm{cocont}} \subseteq \operatorname{\mathcal{QC}}_{< \lambda }$ of Notation 7.6.6.26. This is a reformulation of Example 7.6.6.32.