  In this paper we investigate the categories of braided objects, algebras and
bialgebras in a given monoidal category, some pairs of adjoint functors between
them and their relations. In particular we construct a braided primitive
functor and its left adjoint, the braided tensor bialgebra functor, from the
category of braided objects to the one of braided bialgebras. The latter is
obtained by a specific elaborated construction introducing a braided tensor
algebra functor as a left adjoint of the forgetful functor from the category of
braided algebras to the one of braided objects. The behaviour of these functors
in the case when the base category is braided is also considered.
