  We consider a generalization of (pro)algebraic loops defined on general
categories of algebras and the dual notion of a coloop bialgebra suitable to
represent them as functors. Our main result is the proof that the natural loop
of formal diffeomorphisms with associative coefficients is proalgebraic, and
give a full description of the codivisions on its coloop bialgebra.This result
provides a generalization of the Lagrange inversion formula to series with
non-commutative coefficients, and a loop-theoretic explanation to the existence
of the non-commutative F{\`a}a di Bruno Hopf algebra. MSC: 20N05, 14L17, 18D35,
16T30
