Kerodon

Remark 9.2.4.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ be a flat functor. Assume that $\operatorname{\mathcal{D}}$ is locally $n$-truncated for some integer $n$. Then, for every object $C \in \operatorname{\mathcal{C}}$, the Kan complex $\mathscr {F}(C)$ is $n$-truncated. To prove this, we can apply Lemma 9.2.4.17 (possibly after enlarging $\lambda $) to realize $\mathscr {F}$ as a levelwise filtered colimit of representable functors. Since the collection of $n$-truncated Kan complexes is closed under filtered colimits (Variant 9.1.10.3), we may assume without loss of generality that $\mathscr {F}$ is representable by an object $D \in \operatorname{\mathcal{C}}$. In this case, we are reduced to the assertion that the every morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,D )$ is $n$-truncated, which follows from our assumption on $\operatorname{\mathcal{C}}$. See Proposition 9.2.3.13 for a closely related statement.