The groupoid completion of $\mathcal{C}$[1] is the pair $\webleft (\mathrm{K}_{0}\webleft (\mathcal{C}\webright ),\iota _{\mathcal{C}}\webright )$ consisting of
- A groupoid $\mathrm{K}_{0}\webleft (\mathcal{C}\webright )$;
- A functor $\iota _{\mathcal{C}}\colon \mathcal{C}\to \mathrm{K}_{0}\webleft (\mathcal{C}\webright )$;
- Given another such pair $\webleft (\mathcal{G},i\webright )$, there exists a unique functor $\mathrm{K}_{0}\webleft (\mathcal{C}\webright )\overset {\exists !}{\to }\mathcal{G}$ making the diagram
commute.
