Theorem 9.5.1.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is accessible and cocomplete. Then a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable if and only if it preserves small limits.
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Proof of Theorem 9.5.1.8. Let $\operatorname{\mathcal{C}}$ be an accessible cocomplete $\infty $-category and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ be a continuous functor; we wish to show that $\mathscr {F}$ is representable. Choose a small regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated. The desired result now follows by applying Proposition 9.5.1.11 in the special case $\lambda = \mu = \Omega $, where $ \Omega $ is a strongly inaccessible cardinal. $\square$