[Special characters omitted.] -structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is extension of a [Special characters omitted.] -algebra by an associative algebra. Actions of [Special characters omitted.] -algebras on associative and [Special characters omitted.] -algebras are analyzed, extensions of [Special characters omitted.] -algebras by associative and [Special characters omitted.] -algebras, that they act upon, are constructed. The resulting [Special characters omitted.] -algebra on the deformation complex of a morphism is shown to be quasi-isomorphic to the [Special characters omitted.] -algebra on deformation complex of the corresponding diagram algebra. [Special characters omitted.] -structure is described on the Hochschild complex of a morphism.