Variant 9.1.3.6. Let $\kappa $ be an infinite cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is $\kappa $-filtered. Then $\operatorname{\mathcal{E}}$ is $\kappa $-filtered if and only if it is weakly contractible.
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Proof of Variant 9.1.3.6. Assume that $\operatorname{\mathcal{E}}$ is weakly contractible; we will show that it is $\kappa $-filtered (the converse follows from Proposition 9.1.1.13). If $\kappa = \aleph _0$, this follows from Theorem 9.1.3.2. We may therefore assume without loss of generality that $\kappa $ is uncountable. Fix a diagram $e: K \rightarrow \operatorname{\mathcal{E}}$, where $K$ is a $\kappa $-small simplicial set; we wish to show that there exists a natural transformation from $e$ to a constant diagram. Using Lemma 9.1.3.5, we can reduce to the case where $K$ is weakly contractible. Using Lemma 9.1.3.1, we can reduce further to the situation where $e$ factors through the Kan complex $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ for some object $C \in \operatorname{\mathcal{C}}$. In this case, our assumption that $K$ is weakly contractible guarantees that $e$ is nullhomotopic (when regarded as a diagram $\operatorname{\mathcal{E}}_{C}$), and therefore isomorphic to a constant diagram (when regarded as a diagram in $\operatorname{\mathcal{E}}$). $\square$