  We show that two flat commutative Hopf algebroids are Morita equivalent if
and only if they are weakly equivalent and if and only if there exists a
principal bibundle connecting them. This gives a positive answer to a
conjecture due to Hovey and Strickland. We also prove that principal (left)
bundles lead to a bicategory together with a 2-functor from flat Hopf
algebroids to trivial principal bundles. This turns out to be the universal
solution for 2-functors which send weak equivalences to invertible 1-cells. Our
approach can be seen as an algebraic counterpart to Lie groupoid Morita theory.
