Kerodon

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Corollary 5.6.2.10. Let $\operatorname{\mathcal{Q}}$ be a full subcategory of $\operatorname{\mathcal{QC}}$ and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets having the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is equivalent to an $\infty $-category which belongs to $\operatorname{\mathcal{Q}}$. Then there exists a morphism $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{Q}}\subseteq \operatorname{\mathcal{QC}}$ which is a covariant transport representation of $U$.

Proof. For each vertex $C \in \operatorname{\mathcal{C}}$, choose an $\infty $-category $\mathscr {F}'(C) \in \operatorname{\mathcal{Q}}$ which is equivalent to the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. The construction $C \mapsto \mathscr {F}'(C)$ determines a morphism of simplicial sets $\mathscr {F}': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{Q}}$, where $\operatorname{\mathcal{C}}' = \operatorname{sk}_0(\operatorname{\mathcal{C}})$ is the $0$-skeleton of $\operatorname{\mathcal{C}}$, which is a covariant transport representation of the projection map $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}'$ (see Example 5.6.2.5). Applying Corollary 5.6.2.9, we can extend $\mathscr {F}'$ to a morphism $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ which is a covariant transport representation of $U$. By construction, the morphism $\mathscr {F}$ takes values in the full subcategory $\operatorname{\mathcal{Q}}\subseteq \operatorname{\mathcal{QC}}$. $\square$