Tambara module

Definition. Let $(\mathbb{A},\otimes ,I)$ be a monoidal category. A Tambara module is a profunctor $T﹕{\mathbb{A}}^{op}\times \mathbb{A}\to \mathbf{Set}$ endowed with transformations $t_l^M:T(X;Y)\to T(M\otimes X,M\otimes Y),$ $t_r^M:T(X;Y)\to T(X\otimes M,Y\otimes M),$ that are natural in both $X$ and $Y$, but also dinatural on $M$. These must moreover satisfy the following axioms:

• $t_l^I=id$ and $t_r^I=id$, unitality;
• $t_l^M⨾t_l^N=t_l^{N\otimes M}$ and $t_r^M⨾t_r^N=t_l^{M\otimes N}$, multiplicativity;
• $t_l^M⨾t_r^N=t_r^N⨾t_l^M$, and compatibility.

References:

• Doubles for Monoidal Categories (Pastro and Street, 2008)