On continuity of accessible functors

We prove that for each locally α-presentable category K there exists a regular cardinal γ such that any α-accessible functor out of K (into another locally α-presentable category) is continuous if and only if it preserves γ-small limits; as a consequence we obtain a new adjoint functor theorem specific to the α-accessible functors out of K. Afterwards we generalize these results to the enriched setting and deduce, among other things, that a small V-category is accessible if and only if it is Cauchy complete.