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