Loop of formal diffeomorphisms and F{\`a}a di Bruno coloop bialgebra

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