  In this paper we revisit the hypothesis needed to define the
"paracomposition" operator, an analogue to the classic pull-back operation in
the low regularity setting, first introduced by S. Alinhac in [3]. More
precisely we do so in two directions. First we drop the diffeomorphism
hypothesis. Secondly we give estimates in global Sobolev and Zygmund spaces.
Thus we fully generalize Bony's classic paralinearasition theorem giving sharp
estimates for composition in Sobolev and Zygmund spaces. In order to prove that
the new class of operations benefits of symbolic calculus properties when
composed by a paradifferential operator, we discuss the pull-back of
pseudodifferential and paradifferential operators which then become Fourier
Integral Operators. In this discussion we show that those Fourier Integral
Operators obtained by pull-back are pseudodifferential or paradifferential
operators if and only if they are pulled-back by a diffeomorphism that is a
change of variable. We give a proof of the change of variables in
paradifferential operators.
  Finally we study the cutoff defining paradifferential operators and it's
stability by successive composition. It is known that the cutoff becomes worse
after each composition, we give a slightly refined version of the cutoffs
proposed by H\"ormander in [14] for which give an optimal estimate on the
support of the cutoff after composition.
