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

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