  We propose a definition of Poisson quasi-Nijenhuis Lie algebroids as a
natural generalization of Poisson quasi-Nijenhuis manifolds and show that any
such Lie algebroid has an associated quasi-Lie bialgebroid. Therefore, also an
associated Courant algebroid is obtained. We introduce the notion of a morphism
of quasi-Lie bialgebroids and of the induced Courant algebroids morphism and
provide some examples of Courant algebroid morphisms. Finally, we use paired
operators to deform doubles of Lie and quasi-Lie bialgebroids and find an
application to generalized complex geometry.
