  We study the embedding property in the category of sorted profinite groups.
We introduce a notion of the sorted embedding property (SEP), analogous to the
embedding property for profinite groups. We show that any sorted profinite
group has a universal SEP-cover. Our proof gives an alternative proof for the
existence of a universal embedding cover of a profinite group. Also our proof
works for any full subcategory of the sorted profinite groups, which is closed
under taking finite quotients, fibre product, and inverse limit.
  We also introduce a weaker notion of finitely sorted embedding property
(FSEP), and it turns out to be equivalent to SEP. The advantage of FSEP is to
be able to be axiomatized in the first order language of sorted complete
systems. Using this, we show that any sorted profinite group having SEP has the
sorted complete system whose theory is $\omega$-stable under the assumption
that the set of sorts is countable. In this case, as a byproduct, we get the
uniqueness of a universal SEP-cover of a sorted profinite group, which
generalizes the uniqueness of an embedding cover of a profinite group.
