$(H,R)$-Lie coalgebras and $(H,R)$-Lie bialgebras

Given an $(H,R)$-Lie coalgebra $\Gamma$, we construct $(H,R_T)$-Lie coalgebra $\Gamma^T$ through a right cocycle $T$, where $(H,R)$ is a triangular Hopf algebra, and prove that there exists a bijection between the set of $(H,R)$-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if $(L,[,],\Delta,R)$ is an $(H,R)$-Lie bialgebra of an ordinary Lie algebra then $(L^T,[,],\Delta_T,R_T)$ is an $(H,R_T)$-Lie bialgebra of an ordinary Lie algebra.