  Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the
study of Koszul rings. In this paper, we continue the enquiry of such pairs,
focusing on the description of the second component, as a follow-up of the
study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove
several equivalent characterizations for them. As applications, in the case of
locally finite $R$-rings, we show that a graded $R$-ring is Koszul if and only
if its left (or right) graded dual coring is Koszul. Finally, for finite graded
posets, we obtain that the respective incidence ring is Koszul if and only if
the incidence coring is so.
