  For any double Poisson algebra, we produce a double Poisson vertex algebra
using the jet algebra construction. We show that this construction is
compatible with the representation functor which associates to any double
Poisson (vertex) algebra and any positive integer a Poisson (vertex) algebra.
We also consider related constructions, such as Poisson reductions and
Hamiltonian reductions, with the aim of comparing the different corresponding
categories. This allows us to provide various interesting examples of double
Poisson vertex algebras, in particular from double quivers.
