  We consider universal off-shell Bethe vectors given in terms of Drinfeld
realization of the algebra $U_q(\widehat{gl}_N)$
[arXiv:math/0610517,arXiv:0711.2819]. We investigate ordering properties of the
product of the transfer matrix and these vectors.
  We derive that these vectors are eigenvectors of the transfer matrix if their
Bethe parameters satisfy the universal Bethe equations [arXiv:math-ph/0512037].
