Lemma 9.1.3.1. Let $\kappa $ be an infinite cardinal, let $K$ be a $\kappa $-small simplicial set and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration. If $\operatorname{\mathcal{C}}$ is $\kappa $-filtered, then every diagram $f: K \rightarrow \operatorname{\mathcal{E}}$ admits a natural transformation $\beta : f \rightarrow f'$ where $U \circ f'$ is constant: that is, $f'$ is a diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, for some object $C \in \operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Since $\operatorname{\mathcal{C}}$ is $\kappa $-filtered, we can choose an object $C \in \operatorname{\mathcal{C}}$ and a morphism $\overline{\beta }: (U \circ e) \rightarrow \underline{C}$ in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Using Theorem 5.2.1.1, we can lift $\overline{\beta }$ to a ($U$-cocartesian) natural transformation $\beta : e \rightarrow e'$, where $e'$ is a diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$. $\square$