  It is argued that the dual transformation of non-Abelian monopoles occurring
in a system with gauge symmetry breaking G -> H is to be defined by setting the
low-energy H system in Higgs phase, so that the dual system is in confinement
phase. The transformation law of the monopoles follows from that of
monopole-vortex mixed configurations in the system (with a large hierarchy of
energy scales, v_1 >> v_2) G -> H -> 0, under an unbroken, exact color-flavor
diagonal symmetry H_{C+F} \sim {\tilde H}. The transformation property among
the regular monopoles characterized by \pi_2(G/H), follows from that among the
non-Abelian vortices with flux quantized according to \pi_1(H), via the
isomorphism \pi_1(G) \sim \pi_1(H) / \pi_2(G/H). Our idea is tested against the
concrete models -- softly-broken {\cal N}=2 supersymmetric SU(N), SO(N) and
USp(2N) theories, with appropriate number of flavors. The results obtained in
the semiclassical regime (at v_1 >> v_2 >> \Lambda) of these models are
consistent with those inferred from the fully quantum-mechanical low-energy
effective action of the systems (at v_1, v_2 \sim \Lambda).
