Kerodon

Remark 9.1.1.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a Morita equivalence of $\infty $-categories (Definition 8.5.6.1) and let $\kappa $ be an infinite cardinal. If $\operatorname{\mathcal{D}}$ is $\kappa $-filtered, then $\operatorname{\mathcal{C}}$ is also $\kappa $-filtered. To prove this, we can replace $\operatorname{\mathcal{C}}$ by its essential image and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is a full subcategory of $\operatorname{\mathcal{D}}$. If $f: K \rightarrow \operatorname{\mathcal{C}}$ is a $\kappa $-small diagram, then our assumption that $\operatorname{\mathcal{D}}$ is $\kappa $-filtered guarantees that there exists a natural transformation $f \rightarrow \underline{Y}$, for some object $Y \in \operatorname{\mathcal{D}}$. Since $F$ is a Morita equivalence, the object $Y$ is a retract of some object $X \in \operatorname{\mathcal{C}}$. It follows that there is also a natural transformation from $f$ to the constant diagram $\underline{X}$.