Homotopy Gerstenhaber algebras are strongly homotopy commutative

We show that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative (shc) algebra in the sense of Stasheff–Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a \(\mathbin {\cup _1}\)-product on the bar construction, the structure map becomes homotopy commutative, so that one obtains an shc algebra in the sense of Munkholm.