The Cartier-Konstant-Milnor-Moore (et al.) theorem for Hopf algebras states that a cocommutative Hopf algebra over $\mathbb{C}$ is isomorphic to a smash product of a universal enveloping algebra of a Lie algebra and a group Hopf algebra. Hence in some sense Hopf algebras manage to interpolate between groups and Lie algebras. As a natural setting for both sides of the Lie correspondence, is there any map known within the realm of Hopf algebras that connects the two?

I have seen examples before where people adjoin the (formal) exponential of a primitive element to an algebra to function as for example a quasitriangular structure or as a way to construct a new algebra (eg. Majid '95 lemma 3.1.3), so I am aware that the exponential of a primitive element forms a grouplike element. I also know that in Lusztig's Introduction to Quantum Groups the pairing between a group algebra on a lie group and the corresponding lie algebra is defined, so there is some sort of duality one can formulate within the language of Hopf algebras. I am looking for any result that instead does something like implement the Lie functor as an endofunctor on a (sub-)category of Hopf algebras.