  In this paper, we develop a Riemann-Hilbert approach to the Cauchy problem
for the two-component modified Camassa-Holm (2-mCH) equation based on its Lax
pair. Further via a series of deformations to the Riemann-Hilbert problem
associated with the Cauchy problem by using the $\bar{\partial}$-generalization
of Deift-Zhou steepest descent method, we obtain the long-time asymptotic
approximations of the solutions of the 2-mCH equation in four space-time
regions. Our results also confirm the soliton resolution conjecture and
asymptotically stability of the $N$-soliton solutions for the 2-mCH equation.
