  Let $(H,R)$ be a quasitriangular weak Hopf algebra over a field $k$. We show
that there is a braided monoidal equivalence between the Yetter-Drinfeld module
category $^H_H\mathscr{YD}$ over $H$ and the category of comodules over some
braided Hopf algebra ${}_RH$ in the category $_H\mathscr{M}$. Based on this
equivalence, we prove that every braided bi-Galois object $A$ over the braided
Hopf algebra ${}_RH$ defines a braided autoequivalence of the category
$^H_H\mathscr{YD}$ if and only if $A$ is quantum commutative. In case $H$ is
semisimple over an algebraically closed field, i.e. the fusion case, then every
braided autoequivalence of $^H_H\mathscr{YD}$ trivializable on $_H\mathscr{M}$
is determined by such a quantum commutative Galois object. The quantum
commutative Galois objects in $_H\mathscr{M}$ form a group measuring the Brauer
group of $(H,R)$ as studied in [20] in the Hopf algebra case.
