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, v1 ≫ v2) G over(→, v1) H over(→, v2) 1, under an unbroken, exact color-flavor diagonal symmetry HC + F ∼ over(H, ̃). The transformation property among the regular monopoles characterized by π2 (G / H), follows from that among the non-Abelian vortices with flux quantized according to π1 (H), via the isomorphism π1 (G) ∼ π1 (H) / π2 (G / H). Our idea is tested against the concrete models-softly-broken N = 2 supersymmetric SU (N), SO (N) and USp (2 N) theories, with appropriate number of flavors. The results obtained in the semiclassical regime (at v1 ≫ v2 ≫ Λ) of these models are consistent with those inferred from the fully quantum-mechanical low-energy effective action of the systems (at v1, v2 ∼ Λ).
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