Proposition 9.5.2.3. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is accessible and $\operatorname{\mathcal{D}}$ is presentable. Then $G$ admits a left adjoint if and only if it is accessible and preserves small limits.
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Proof of Proposition 9.5.2.3. Assume that $G$ is accessible and preserves small limits; we wish to show that $G$ admits a left adjoint (the converse follows from Corollaries 7.1.4.28 and 9.4.7.18). By virtue of Corollary 6.2.6.2, it will suffice to show that for every functor $h_ C: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresentable by an object $C \in \operatorname{\mathcal{C}}$, the composition $(h_{C} \circ G): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by an object of $\operatorname{\mathcal{D}}$. Using the criterion of Theorem 9.5.1.9, we are reduced to showing that $h_{C} \circ G$ is accessible and preserves small limits. This follows from our hypotheses on $G$, since the functor $h_{C}$ is accessible (Example 9.4.7.16) and preserves all limits which exist in $\operatorname{\mathcal{C}}$ (Corollary 7.4.1.23). $\square$