  Skew lattices are non-commutative generalizations of lattices. The coset
structure decomposition is an original approach to the study of these algebras
describing the relation between its rectangular classes. In this paper we will
look at the category determined by these rectangular algebras and the morphisms
between them, showing that not all skew lattices can determine such a category.
Furthermore, we will present a class of examples of skew lattices in rings that
are not strictly categorical, and present sufficient conditions for skew
lattices of matrices in rings to constitute $\wedge$-distributive skew
lattices.
