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The MuCool program undertaken by the US Neutrino Factory and Muon Collider Collaboration is to study the behavior of muon ionization cooling channel components. A single superconducting coupling solenoid magnet is necessary to pursue the research and development work on the performance of high gradient, large size RF cavities immersed in magnetic field, which is one of the main challenges in the practical realization of ionization cooling of muons. The MuCool coupling magnet is to be built using commercial copper based niobium titanium conductors and cooled by two cryo-coolers with each cooling capacity of 1.5 W at 4.2 K. The solenoid magnet will be powered by using a single 300 A power supply through a single pair of binary leads that are designed to carry a maximum current of 210 A. The magnet is to be passively protected by cold diodes and resistors across sections of the coil and by quench back from the 6061 Al mandrel in order to lower the quench voltage and the hot spot temperature. The magnet is currently under construction. This paper presents the updated design and fabrication progress on the MuCool coupling magnet.