In this chapter we prepare the ground to prove that consecutive renormalizations of circle diffeomorphisms converge to the set of rotations. In the previous chapter we described the three ingredients which are responsible for this convergence, namely circle diffeomophisms converge to the set of Möbius circle maps, Möbius circle maps converge to the set of affine circle maps and affine circle maps converge to the set of rotations. Now it is time to merge these three ingredients and show their collaboration. Indeed, these mechanisms are also active when one renormalizes generalized diffeomorphisms. Different from the specific cases of Möbius maps and affine maps where the effects of the ingredients are described by exact formulas, when applying them in the general case of circle diffeomorphisms one gets errors. This is described in Proposition 6.1.3 where one can clearly recognize the ingredients responsible for convergence in the cases of affine maps and the Möbius maps. Fortunately, the errors appearing in the generalized context, decay very fast.