We give an algebraic proof of properness of moduli stacks of gauged maps satisfying a stability conditition introduced by Mundet and Schmitt. The proof combines a git construction of Schmitt, properness for twisted stable maps by Abramovich-Vistoli, a variation of a boundedness argument due to Ciocan-Fontanine-Kim-Maulik, and a removal of singularities for bundles on surfaces in Colliot-Thel`ene-Sansuc.
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