Kerodon

Warning 9.2.5.20. In the formulation of Corollary 9.2.5.19, the assumption that $\operatorname{\mathcal{C}}$ is idempotent complete cannot be omitted. For example, suppose that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor which exhibits $\operatorname{\mathcal{D}}$ as an idempotent completion of $\operatorname{\mathcal{D}}$. Then $F$ is $\kappa $-left exact for every regular cardinal $\kappa $ (Example 9.2.5.14). However, $F$ admits a left adjoint only if it is an equivalence of $\infty $-categories: that is, only if the $\infty $-category $\operatorname{\mathcal{C}}$ is idempotent complete.