Warning 9.5.1.10. In the formulation of Theorem 9.5.1.9, the assumption that the functor $\mathscr {F}$ is accessible cannot be omitted. For example, let $\operatorname{\mathcal{C}}$ be (the nerve of) the category of (small) groups. For every small regular cardinal $\kappa $, choose a simple group $G_{\kappa }$ of cardinality $\kappa $. Then the functor
\[ \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\quad \quad \mathscr {F}(G) = \prod _{\kappa } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( G_{\kappa }, G) \]
preserves small limits but is not corepresentable. This functor is well-defined because the product on the right hand side is always small (since every group homomorphism $G_{\kappa } \rightarrow G$ is trivial when $\kappa $ is larger than the cardinality of $G$).