A Poisson Hopf algebra is both a Poisson algebra and a Hopf algebra such that the comultiplication and

the counit are morphisms of Poisson algebras. Such objects are situated at the border between Poisson

geometry and quantum groups. We investigate the category of Poisson Hopf algebras with respect to

properties such as (co)completeness or existence of (co)free objects. For instance, 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

show that the category of Poisson Hopf algebras is a reflective subcategory of the category of Poisson

bialgebras. Along the way, we point out the differences between the category of Poisson Hopf algebras

and the category of Hopf algebras.

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