  In the preceeding paper we constructed an infinite exact sequence a la
Villamayor-Zelinsky for a symmetric finite tensor category. It consists of
cohomology groups evaluated at three types of coefficients which repeat
periodically. In the present paper we interpret the middle cohomology group in
the second level of the sequence. We introduce the notion of coring categories
and we obtain that the mentioned middle cohomology group is isomorphic to the
group of Azumaya quasi coring categories. This result is a categorical
generalization of the classical Crossed Product Theorem, which relates the
relative Brauer group and the second Galois cohomology group with respect to a
Galois field extension. We construct the colimit over symmetric finite tensor
categories of the relative groups of Azumaya quasi coring categories and the
full group of Azumaya quasi coring categories over $vec$. We prove that the
latter two groups are isomorphic.
