  The coproduct of a Feynman diagram is set up through identifying the
perturbative unitarity of the S-matrix with the cutting equation from the
cutting rules. On the one hand, it includes all partitions of the vertex set of
the Feynman diagram and leads to the circling rules for the largest time
equation. Its antipode is the conjugation of the Feynman diagram. On the other
hand, it is regarded as the integration of incoming and outgoing particles over
the on-shell momentum space. This causes the cutting rules for the cutting
equation. Its antipode is an advanced function vanishing in retarded regions.
Both types of coproduct are well-defined for a renormalized Feynman diagram
since they are compatible with the Connes--Kreimer Hopf algebra.
