G∞G∞-structure on the deformation complex of a morphism

G∞G∞-structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is the extension of a B∞B∞-algebra by an associative algebra. Actions of B∞B∞-algebras on associative and B∞B∞-algebras are analyzed; extensions of B∞B∞-algebras by associative and B∞B∞-algebras that they act upon are constructed. The resulting G∞G∞-algebra on the deformation complex of a morphism is shown to be quasi-isomorphic to the G∞G∞-algebra on the deformation complex of the corresponding diagram algebra.