  A survey of properties of the adjunction involving a semisymmetrization
functor, which was suggested by J.D.H. Smith, and which maps the category of
quasigroups with homotopies to the category of semisymmetric quasigroups with
homomorphisms, is given. A new semisymmetrization functor is suggested. This
functor maps a quasigroup to its square instead to its cube as it was the case
with the former functor.
