Semirings from traces of categories

Question

Let $\mathcal C$ be a small category. Its trace $\operatorname{Tr}(\mathcal C)$ is defined as the coend of the hom functor. We can identify $\operatorname{Tr}(\mathcal C)$ with quotient of $\coprod_{X \in \mathcal C} \operatorname{End}(X)$ under relations $fg \sim gf$. In case of category $\bf{Fin}$ of finite sets, nLab notes that since products distributes under coproducts, the operations $\operatorname{Hom}(X, X) \times \operatorname{Hom}(Y, Y) \rightarrow \operatorname{Hom}(X\times Y, X\times Y)$ and $\operatorname{Hom}(X, X) \times \operatorname{Hom}(Y, Y) \rightarrow \operatorname{Hom}(X\amalg Y, X\amalg Y)$ indice a rig structure on $\operatorname{Tr}(\bf{Fin})$, which coincides with the Burnside rig of $\mathbb Z$, but sadly the page lacks additional references. Do we any interesting rig structure with different categories where products and coproducts are distributive? What is known about properties of such rigs? Can this construction be made functorial?