A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group $mathrm U_q(mathcal L(mathfrak{sl}_3))$ is given. The full proof of the functional relations in the form independent of
the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described.
We discuss the main points of the quantum group approach in the theory of quantum integrable systems and illustrate them for the case of the quantum group $U_q(mathcal L(mathfrak{sl}_2))$. We give a complete set of the functional relations correcting
inexactitudes of the previous considerations. A special attention is given to the connection of the representations used to construct the universal transfer operators and $Q$-operators.
We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group $U_q(mathcal L(m
athfrak{sl}_2))$ related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.
