The duality theory for monotone interacting particle systems was initiated by Gray (1986) and further developed by Sturm and Swart (2018). It contains the better known additive duality as a special case but differs in the sense that the dual process contains not only single particles but also pairs, triples, and general $n$-tuples of particles, which correspond to the fact that in the forward process sometimes several particles are needed to create one particle at a later time. In earlier work, the dual process was constructed for finite initial states only, but, assuming that the empty state is a trap for the forward process, we show that the dual process can be started in infinite initial states and has an upper invariant law. It can therefore be viewed as some sort of interacting particle system in its own right. For the monotone dual of a cooperative contact process, we show that the upper invariant law is the long-time limit started from any nontrivial homogeneous invariant law. We use this to prove continuity of the survival probability of the forward process as a function of its parameters.