  Let F be the free group functor, left adjoint to the forgetful functor
between the category of groups GRP and the category of sets SET. Let f from A
to B, and h from A to C be two functions in SET and let Ker(F(f)) and Ker(F(h))
be the kernels of the induced morphisms between free groups. Provided that the
kernel pairs Eq(f) and Eq(h) of f and h permute (such as it is the case when
the pushout of f and h is a double extension in SET), this short article
describes a method to rewrite a general element in the intersection of
Ker(F(f)) and Ker(F(g)) as a product of generators in A which is
(f,h)-symmetric in the sense of the higher covering theory of racks and
quandles.
