  In this note we observe that any compact quantum group monoidally equivalent,
in a nice way, to a free wreath product of a compact quantum group $G$ by the
quantum automorphism group of a finite dimensional C*-algebra with a
$\delta$-form is actually isomorphic to a free wreath product of $G$ by the
quantum automorphism group of another finite dimensional C*-algebra with a
$\delta$-form.
