  In the theory of the moduli-stacks of n-pointed stable curves, there are two
fundamental functors, contraction and stabilization. These functors are
constructed in [4], where they are used to show that the various
\bar{M_{g,n}}'s are DM-stacks. We give here a more elementary and more complete
proof of the fact that stabilization, as it is defined in [4], is a functor.
The fact that contraction is a functor inverse to the stabilization-functor we
think is satisfactorily treated in [4]. In a very recent publication [1] there
is a proof based on the proof in [4], but it is adapted to the case of moduli
stacks over the complex numbers. In this note we work in the most general
setting, i.e. over the integers.
