Proposition 9.5.2.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is presentable and $\operatorname{\mathcal{D}}$ is locally small. Then $F$ admits a right adjoint if and only if it is cocontinuous: that is, it preserves small colimits.
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Proof. Assume that $F$ is cocontinuous; we will show that it admits a right adjoint (the reverse implication follows from Corollary 7.1.4.28). By virtue of Corollary 6.2.6.2, it will suffice to show that for every functor $h_{D}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ which is representable by an object $D \in \operatorname{\mathcal{D}}$, the composition $(h_{D} \circ F^{\operatorname{op}}): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable by an object of $\operatorname{\mathcal{C}}$. Using Theorem 9.5.1.8, we are reduced to showing that the functor $h_{D} \circ F^{\operatorname{op}}$ preserves small limits. This follows from the cocontinuity of $F$, since $h_{D}$ preserves all limits which exist in $\operatorname{\mathcal{D}}^{\operatorname{op}}$ (Corollary 7.4.1.23). $\square$