Bethe vectors of quantum integrable models based on $U_q(\hat{\mathfrak{gl}}_N)$

We study quantum $U_q(\widehat{\mathfrak {gl}}_{N})$ integrable models solvable by the nested algebraic Bethe ansatz. Different formulas are given for the right and left universal off-shell nested Bethe vectors. It is shown that these formulas can be related by certain morphisms of the positive Borel subalgebra in $U_q(\widehat{\mathfrak {gl}}_{N})$ into analogous subalgebra in $U_{q^{-1}}(\widehat{\mathfrak {gl}}_N)$.