Let $A$ be a honest algebra or more generally, a DG algebra. It is known that the Hochschild cochain complex is quasi-isomorphic to the derived Hom complex, i.e. one has $$\mathrm{HH}^{\bullet}(A,\,A)\,\cong\,\mathrm{H}_{-\bullet}[\mathrm{RHom}_{A^e}(A,\,A)]\,.$$ Moreover, there is a Gerstanharber bracket structure on $\mathrm{HH}^{\bullet}(A,\,A)$.
My question is that by the isomorphism above, can we construct this bracket from the derived Hom complex? Or is the Gerstanharber bracket related to some operations in the derived category of $A^e$?
Thanks in advance.
