On Continuity of Accessible Functors

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