Partial Tambara structure on the Burnside biset functor, induced from a derivator-like system of adjoint triplets

In the previous article “A Mackey-functor theoretic interpretation of biset functors  ”, we have constructed the 2-category SSof finite sets with variable finite group actions  , in which bicoproducts and bipullbacks exist. As shown in it, biset functors can be regarded as a special class of Mackey functors on SS.In this article, we equip SS with a system of adjoint triplets, which satisfies properties analogous to a derivator. This system encodes the six operations for finite groups. As a corollary, we show that the associated Burnside rings satisfy analogous properties to a Tambara functor, in the context of biset functor theory.