We prove that for each locally alpha-presentable category K there exists a regular cardinal gamma such that any alpha-accessible functor out of K (into another locally alpha-presentable category) is continuous if and only if it preserves gamma-small limits; as a consequence we obtain a new adjoint functor theorem specific to the alpha-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.
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