Given a finite abelian category $\mathcal{C}$, we can associate to $\mathcal{C}$ its Grothendieck group $\mathsf{Gr}(\mathcal{C})$, which is the free abelian group generated by isomorphism classes of simple objects in $\mathcal{C}$.
Now, let $\mathcal{R}$ be a $\mathcal{V}$-enriched bigroupoid, where $\mathcal{V}=(\mathsf{Cat}_{ab}^{fin},\boxtimes)$ is the monoidal category of finite abelian categories (probably with Deligne product). Is there a higher analogue of the Grothendieck group for this situation? Something along the lines that, shrinking down the morphism categories to "iso classes" (whatever these are) the composition functor defines a groupoid structure on them?
If this is not the case, is there a setting in which the term "Grothendieck groupoid" makes sense?
