  In the previous papers we found a direct method to confirm, for any square
matrix, if it is associated to any categories or not. According to this method,
the matrix 2 (all coefficients are 2) of a given order, admits associated
categories. We determine in this paper the isomorphism classes of reduced
categories that are associated with the matrix 2 of order 1, 2 and 3. More
generally, we managed to give upper and lower bounds for the cardinal of the
set of all reduced categories associated to the matrix 2 of a given order.
