  Let p be a prime, P a finite p-group and F a Frobenius P-category. The
question on the existence of a suitable category Lsc extending the full
subcategory of F over the set of F-selfcentralizing subgroups of P goes back to
Dave Benson in 1994. In 2002 Carles Broto, Ran Levi and Bob Oliver formulate
the existence and the uniqueness of the category Lsc in terms of the annulation
of an obstruction 3-cohomology element and of the vanishing of a 2-cohomology
group, and they state a sufficient condition for the vanishing of these
n-cohomology groups. Recently, Amy Chermak has proved the existence and the
uniqueness of Lsc via his objective partial groups, and Bob Oliver, following
some of Chermak's methods, has also proved the vanishing of those n-cohomology
groups for n > 1, both applying the Classification of the finite simple groups.
Here we give direct proofs of the existence and the uniqueness of Lsc;
moreover, in [11] we already show that Lsc can be completed in a suitable
category L extending F and here we prove some functoriality of this
correspondence.
