^{x}L}}$ up to $(H\cap {\ ^{x}L})$-conjugacy, is an $R$-basis of $\mu_{R}(G)$.

Section $3$ of [@tw].

\[prop\_b\] The Mackey algebra $\mu_{R}(G)$ is isomorphic to $RB(\Omega_{G}^{2})$, where $\Omega_{G}$ is the $G$-set: $\sqcup_{L\leqslant G} G/L$.

The proof can be found in Proposition $4.5.1$ of [@bouc_green]. Let us recall that an explicit isomorphism $\beta$ can be defined on the generators of $\mu_{R}(G)$ by $\beta(t^{K}_{H}):=$ $$\xymatrix{
& G/H\ar[dl]_{\pi_{H}^{K}}\ar@{=}[dr]& \\
G/K & & G/H
}$$ where $\pi_{H}^{K} : G/H\to G/K$ is the canonical map. Similarly, we define $\beta(r^{K}_{H}):=$ $$\xymatrix{
& G/H\ar[dr]^{\pi_{H}^{K}}\ar@{=}[dl]& \\
G/H & & G/K
}$$ and $\beta(c_{g,H}):=$ $$\xymatrix{
& G/{^{g}H}\ar[dr]^{\gamma_{g,H}}\ar@{=}[dl]& \\
G/{^{g}H} & & G/H
}$$ where $\gamma_{g,H}(x{\ ^{g}H})=xgH$. One can check that this gives an isomorphism of algebras.

There is an equivalence of categories

$Mack_{R}(G)\cong \mu_{R}(G)$-Mod.

Burnside Trace.
---------------

There is a tensor product in the category of Mackey functors (see [@bouc_green], e.g.). With this tensor product, the category is a closed symmetric monoidal category with the Burnside functor as unit. So, using the formalism of May ([@may_trace]) where the dualizable Mackey functors are exactly the finitely generated projective Mackey functors, Bouc has defined the notion of Burnside dimension and Burnside trace for these Mackey functors ([@bouc_burnside_dim]). Let $M$ be a finitely generated projective Mackey functor. The Burnside trace, denoted by $Btr$ is a map from $End_{Mack_{R}(G)}(M)$ to $RB(G)$. Let $RB_{X}$ be the Dress construction of the Burnside functor at the finite $G$-set $X$ (see [@dress] or [@bouc_green]). It is well known that $RB_{X}$ is a finitely generated projective Mackey functor. By an adjunction property, we have an isomorphism of $R$-algebras $End_{Mack_{R}(G)}(RB_{X})\cong RB(X^2)$ where the product on this ring is defined as in Example \[burnside\]. Using these identifications, the Burnside trace on this Mackey fu