  A class of left bialgebroids whose underlying algebra $A\sharp H$ is a smash
product of a bialgebra $H$ with a braided commutative Yetter--Drinfeld
$H$-algebra $A$ has recently been studied in relation to models of field
theories on noncommutative spaces. In [A. Borowiec, A. Pachol, ``Twisted
bialgebroids versus bialgebroids from a Drinfeld twist'', J. Phys. A50 (2017)
055205] a proof has been presented that the bialgebroid $A_F\sharp H^F$ where
$H^F$ and $A_F$ are the twists of $H$ and $A$ by a Drinfeld 2-cocycle $F = \sum
F^1\otimes F^2$ is isomorphic to the twist of the bialgebroid $A\sharp H$ by
the bialgebroid 2-cocycle $\sum 1\sharp F^1\otimes 1\sharp F^2$ induced by $F$.
They assume $H$ is quasitriangular, which is reasonable for many physical
applications. However the proof and the entire paper take for granted that the
coaction and the prebraiding are both given by special formulas involving the
R-matrix. There are counterexamples of Yetter--Drinfeld modules over
quasitriangular Hopf algebras which are not of this special form. Nevertheless,
the main result essentially survives. We present a proof with general coaction
and the correct prebraiding, and even without the assumption of
quasitriangularity.
