The article investigates the fine structure of free cocommutative Hopf algebras, which are unshuffle Hopf algebras, that is enveloping algebras of free Lie algebras (possibly up to completion), and of the dual commutative cofree Hopf algebras, which are shuffle Hopf algebras. We introduce and study in particular base changes related to Lie idempotents families. The article also develops a categorical and duality framework to address the non graded case that includes for example quasi-shuffle Hopf algebras. Lastly, we survey various classical examples. We develop in detail the one of finite topologies that illustrates how one can take advantage of various extra algebraic structures such as infinitesimal bialgebras or double bialgebras structures in this context.