We extend Kontsevich's formality morphism to a homotopy braces morphism and to a homotopy Gerstenhaber morphism. We show that this morphism is homotopic to 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, that is, 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.
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