It is argued that the dual transformation of non-Abelian monopoles occurring in a system with gauge symmetry breaking G \longrightarrow 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 \gg v_2) G {\stackrel {v_1} {\longrightarrow}} H {\stackrel {v_2} {\longrightarrow}} \emptyset, 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) \over \pi_2(G/H)}.