Theorem 9.5.1.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is accessible and complete. Then a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable if and only if it is accessible and preserves small limits.
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Proof of Theorem 9.5.1.9. Let $\operatorname{\mathcal{C}}$ be an accessible complete $\infty $-category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor which is continuous and accessible; we wish to show that $\mathscr {F}$ is corepresentable by an object of $\operatorname{\mathcal{C}}$ (the reverse implication follows from Corollary 7.4.1.23 and Remark 9.4.6.8). Fix a small regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is $\kappa $-accessible and the functor $\mathscr {F}$ is $\kappa $-finitary. The desired result now follows by applying Proposition 9.5.1.13 in the special case where $\lambda = \mu = \Omega $ is a strongly inaccessible cardinal. $\square$