Formal groups and unipotent affine groups in non-categorical symmetry

As is well known, in characteristic zero, the Lie algebra functor gives two category equivalences, one from the formal groups to the finite-dimensional Lie algebras, and the other from the unipotent algebraic affine groups to the finite-dimensional nilpotent Lie algebras. We prove these category equivalences in a quite generalized framework, proposed by Gurevich [D.I. Gurevich, The Yang–Baxter equation and generalization of formal Lie theory, Soviet Math. Dokl. 33 (1986) 758–762] and later by Takeuchi [M. Takeuchi, Survey of braided Hopf algebras, in: N. Andruskiewitsch, et al. (Eds.), New Trends in Hopf Algebra Theory, in: Contemp. Math., vol. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 301–324], of vector spaces with non-categorical symmetry. We remove the finiteness restriction from the objects, by using the terms of Hopf algebras and Lie coalgebras.