  In this paper we use duality techniques to study a combination of the
well-known contact process (CP) and the somewhat less-known annihilating
branching process. As the latter can be seen as a cancellative version of the
contact process, we rebrand it as the cancellative contact process (cCP). Our
process of interest will consist of two entries, the first being a CP and the
second being a cCP. We call this process the double contact process (2CP) and
prove that it has (depending on the model parameters) at most one invariant law
under which ones are present in both processes. In particular, we can choose
the model parameter in such a way that CP and cCP are monotonely coupled. In
this case also the above mentioned invariant law will have the property that,
under it, ones in the cCP can only be present at sites where there are also
ones in the CP. Along the way we extend the dualities for Markov processes
discovered in our paper "Commutative monoid duality" to processes on infinite
state spaces so that they, in particular, can be used for interacting particle
systems.
