Deligne-Knop tensor categories and functoriality

A general construction of Knop creates a symmetric monoidal category T(A,δ) from any regular category A and a fixed degree function δ. A special case of this construction are the Deligne categories Rep(Sₜ) and Rep(GLₜ(Fq)). We discuss when a functor F:A → A' between regular categories induces a symmetric monoidal functor T(A,δ) → T(A',δ'). We then give a criterion when a pair of adjoint functors between two regular categories A, \ A' lifts to a pair of adjoint functors between T(A,δ) and T(A',δ').