  The biquantization of symmetric pairs was studied in Cattaneo and Torossian
in terms of Kontsevich-like graphs. This paper, also in view of recent results
in Calaque et al, amends a minor mistake that did not spoil the main results of
the paper. The mistake consisted in ignoring a regular term in the boundary
contribution of some propagators. On the other hand, its correction brings back
the quantum shift, present in the approaches by the orbit method, that was
otherwise puzzlingly missing. In addition a detailed comparison of the two,
equivalent, ways of defining biquantization working on the upper half plane or
on one quadrant is presented, as well as a more conceptual approach to
biquantization and the due corrections of some results of Cattaneo and
Torossian in view of the aforementioned correction by the quantum shift.
Finally, we review the triquantization construction developed for the treatment
of characters by taking into accounts the same boundary problem as for the
biquantization.
