for any homogeneous elements x, y and z in L. Notice here that the differential lowers the degree and so this differential graded Lie algebra is considered to be homologically graded.
A morphism of differential graded Lie algebras is a graded linear map f:L\to L^\prime that commutes with the bracket and the differential, i.e. f [x,y]_{L}=[f(x),f(y)]_{L^\prime} and f (d_L x) = d_{L^\prime} f (x) .