  In our previous paper, we discussed the hyperbolization of the configuration
space of n(> 4) marked points with weights in the projective line up to
projective transformations. A variation of the weights induces a deformation.
It was shown that this correspondence of the set of the weights to the
Teichm\"uller space when n = 5 and to the Dehn filling space when n= 6 is
locally one-to-one near the equal weight. In this paper, we establish its
global injectivity.
