  The notion of quasi-Poisson $G$-spaces with $D/G$-valued moment maps was
introduced by Alekseev and Kosmann-Schwarzbach in 1999. Our main result is a
\emph{Lifting Theorem}, establishing a bijective correspondence between the
categories of quasi-Poisson $G$-spaces with $D/G$-valued moment maps and of
quasi-Poisson $G\times G$-spaces with $D$-valued moment maps. Using this
result, we give simple constructions of fusion and conjugation for these
spaces, and new examples coming from moduli spaces.
