  We extend M. Kontsevich's formality morphism to a homotopy braces morphism
and to a homotopy Gerstenhaber morphism. We show that this morphism is
homotopic to D. Tamarkin's formality morphism, obtained using formality of the
little disks operad, if in the latter construction one uses the
Alekseev-Torossian associator. Similar statements can also be shown in the
"chains" case, i.e., on Hochschild homology instead of cohomology. This settles
two well known and long standing problems in deformation quantization and
unifies the several known graphical constructions of formality morphisms and
homotopies by Kontsevich, Shoikhet, Calaque, Rossi, Alm, Cattaneo, Felder and
the author.
