  We construct the analogue of Takeuchi's free Hopf algebra in the setting of
Poisson Hopf algebras. More precisely, we prove that there exists a free
Poisson Hopf algebra on any coalgebra or, equivalently that the forgetful
functor from the category of Poisson Hopf algebras to the category of
coalgebras has a left adjoint. In particular, we also prove that the category
of Poisson Hopf algebras is a reflective subcategory of the category of Poisson
bialgebras. Along the way, we describe coproducts and coequalizers in the
category of Poisson Hopf algebras, therefore showing that the latter category
is cocomplete.
