  E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter [J. Knot theory Ramifications
(2010)] pointed out that by considering inner automorphism groups of quandles,
one have a functor from the category of quandles with surjective homomorphisms
to that of groups with surjective homomorphisms. In this paper, we focus on
faithful quandles. As main results, we give category equivalence between the
category of faithful quandles with surjective quandle homomorphisms and that of
pairs of groups and their generators with suitable group homomorphisms. We are
also interested in injective quandle homomorphisms. By defining suitable
morphisms among pairs of groups and their generators, we obtain a category
which is equivalent to the category of faithful quandles with injective quandle
homomorphisms.
