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A general construction of Knop creates a symmetric monoidal category T (A, ) from any regular category A and a fixed degree function. 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, ) T (A', '). 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, ) and T (A', ').
Entova-Aizenbud et al. (Thu,) studied this question.
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