  Any $\Gamma$-graded categorical group is determined by a factor set of a
categorical group. This paper studies the factor set of the group $\Gamma$ with
coefficients in the categorical group of the type $(\Pi,A).$ Then, an
interpretation of the notion of $\Gamma-$operator $3-$cocycle is presented and
the proof of cohomological classification theorem for the a $\Gamma-$graded
Gr-category is also presented.
