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A Josephson junction is made of two superconductors sandwiching an insulator, and a Josephson vortex is a magnetic vortex (flux tube) absorbed into the Josephson junction, whose dynamics can be described by the sine-Gordon equation. In a field theory framework, a flexible Josephson junction was proposed, in which the Josephson junction is represented by a domain wall separating two condensations and a Josephson vortex is a sine-Gordon soliton in the domain wall effective theory. In this paper, we propose a Josephson junction of non-Abelian color superconductors and show that a non-Abelian vortex (color magnetic flux tube) absorbed into it is a non-Abelian Josephson vortex represented as a non-Abelian sine-Gordon soliton in the domain wall effective theory, that is the U(N) principal chiral model.