Quantized higher Teichmüller theory, as described by Fock and Goncharov, assigns an algebra and its representation to a surface and a Lie group. This assignment is equivariant with respect to the action of the mapping class group of the surface, and is conjectured to give an analog of a modular functor, that is it should respect the operation of cutting and gluing of surfaces. In this talk I will outline a proof of the above conjecture, and explain how it is related to representation theory of quantum groups. This talk will be mostly based on joint works with Gus Schrader.