A covariant Stinespring theorem

We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be group, and let T:= Rep(G) the rigid C*-tensor category of continuous unitary representations G. Mod(T) C*-2-category cofinite semisimple finitely decomposable T-module categories. show that G-C*-algebras can identified with equivalence classes 1-morphisms out object T in Mod(T). For X: -> M1, Y: M2, we completely positive maps between corresponding 'dilated' to isometries t: X Y \otimes E, where E: M2 M1 is some 'environment' 1-morphism. Dilations are unique up partial isometry on environment; particular, dilation minimising dimension environment unitary. When group this recovers previous Stinespring-type theorems.