  We prove that for each locally $\alpha$-presentable category $\mathcal K$
there exists a regular cardinal $\gamma$ such that any $\alpha$-accessible
functor out of $\mathcal K$ (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 $\mathcal K$. Afterwards we generalize
these results to the enriched setting and deduce, among other things, that a
small $\mathcal V$-category is accessible if and only if it is Cauchy complete.
