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In the previous article 'A Mackey-functor theoretic interpretation of bisetfunctors', we have constructed the 2-category $\mathbb{S}$ of finite sets withvariable finite group actions, in which bicoproducts and bipullbacks exist. Asshown in it, biset functors can be regarded as a special class of Mackeyfunctors on $\mathbb{S}$.  In this article, we equip $\mathbb{S}$ with a system of adjoint triplets,which satisfies properties analogous to a derivator. This system encodes thesix operations for finite groups. As a corollary, we show that the associatedBurnside rings satify analogous properties to a Tambara functor, in the contextof biset functor theory.

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