  We prove a Lie 2-group torsor version of the well-known one-one
correspondence between fibered categories and pseudofunctors. Consequently, we
obtain a weak version of the principal Lie group bundle over a Lie groupoid.
The correspondence also enables us to extend a particular class of principal
2-bundles to be defined over differentiable stacks. We show that the
differential geometric connection structures introduced in the authors'
previous work, combine nicely with the underlying fibration structure of a
principal 2-bundle over a Lie groupoid. This interrelation allows us to derive
a notion of parallel transport in the framework of principal 2-bundles over Lie
groupoids along a particular class of Haefliger paths. The corresponding
parallel transport functor is shown to be smooth. We apply our results to
examine the parallel transport on an associated VB-groupoid.
