Towards Commutator theory for relations. IV
  We provide more characterizations of varieties having a term Mal'cev modulo two functions $F$ and $G$. We characterize varieties neutral in the sense of $F$, that is varieties satisfying $R \subseteq F(R)$. We present examples of global operators satisfying the homomorphism property, in particular we show that many known commutators satisfy the homomorphism property. 
Paolo Lipparini