  In this paper, we consider the (crystalline) prismatic crystals on a scheme
$\mathfrak{X}$. We classify the crystals by $p$-connections on a certain ring
and prove a cohomological comparison theorem. This equivalence is more general
than the Ogus' equivalence. Moreover, we will prove the equivalence between the
crystalline crystals and prismatic crystals by gluing local Ogus-Vologodsky
correspondence.
