Abstract
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.
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