Corollary 9.2.5.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\kappa $ be a regular cardinal. Assume that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small, that $\operatorname{\mathcal{D}}$ is locally $\kappa $-small, and that $\operatorname{\mathcal{C}}$ is idempotent complete. Then $F$ admits a left adjoint $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ if and only if it is $\kappa $-left exact.
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Proof. Assume that $F$ is $\kappa $-left exact; we will show that it admits a left adjoint (the converse is a special case of Corollary 9.2.5.18). By virtue of Corollary 6.2.6.2, it will suffice to show that for each object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ has an initial object. Our assumption that $F$ is $\kappa $-left exact guarantees that $\widetilde{\operatorname{\mathcal{C}}}$ is $\kappa $-cofiltered (Theorem 9.2.5.15), and our assumption that $\operatorname{\mathcal{C}}$ is idempotent complete guarantees that $\widetilde{\operatorname{\mathcal{C}}}$ is also idempotent complete (Corollary 8.5.4.24). It will therefore suffice to show that the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ is essentially $\kappa $-small (Proposition 9.1.8.10). The $\infty $-category $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small by assumption, and the assumption that $\operatorname{\mathcal{D}}$ is locally $\kappa $-small guarantees that projection map $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ is an essentially $\kappa $-small left fibration (Corollary 5.6.7.7). The desired result now follows from Corollary 4.7.9.12. $\square$