Kerodon

Example 9.2.5.14 (Morita Equivalence). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a Morita equivalence of $\infty $-categories (see Definition 8.5.6.1). Then $F$ is $\kappa $-left exact for every regular cardinal $\kappa $. To prove this, choose a left fibration $\widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$, where the $\infty $-category $\widetilde{\operatorname{\mathcal{D}}}$ is $\kappa $-cofiltered; we wish to show that the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$ is also $\kappa $-cofiltered. This follows from Remark 9.1.1.13, since the projection map $\widetilde{F}: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \widetilde{\operatorname{\mathcal{D}}}$ is also a Morita equivalence (Corollary 8.5.6.15).