Corollary 9.1.3.3. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration between essentially $\kappa $-small $\infty $-categories, and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a covariant transport representation for $U$ (Definition 5.6.5.1). Suppose that $\operatorname{\mathcal{C}}$ is filtered. Then $\operatorname{\mathcal{E}}$ is filtered if and only if the colimit $\varinjlim ( \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \in \operatorname{\mathcal{S}}^{< \kappa }$ is contractible.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Proposition 7.4.3.1, the colimit $\varinjlim ( \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} )$ is contractible if and only if the $\infty $-category $\operatorname{\mathcal{E}}$ is weakly contractible. The desired result is therefore a reformulation of Theorem 9.1.3.2. $\square$