The additive completion of the biset category

Let R be a commutative unital ring. We construct a category CR of fractionsX/G, where G is a finite group and X is a finite G-set, and with morphisms given by R-linear combinations of spans of bisets. This category is an additive, symmetric monoidal and self-dual category, with a Krull-Schmidt decomposition for objects. We show that CR is equivalent to the additive completion of the biset category and that the category of biset functors over R is equivalent to the category of R-linear functors from CR to R-Mod. We also show that the restriction of one of these functors to a certain subcategory of CR is a fused Mackey functor.