The core of $\mathcal{C}$ is the pair $\webleft (\mathsf{Core}\webleft (\mathcal{C}\webright ),\iota _{\mathcal{C}}\webright )$ consisting of
- A groupoid $\mathsf{Core}\webleft (\mathcal{C}\webright )$;
- A functor $\iota _{\mathcal{C}}\colon \mathsf{Core}\webleft (\mathcal{C}\webright )\hookrightarrow \mathcal{C}$;
- Given another such pair $\webleft (\mathcal{G},i\webright )$, there exists a unique functor $\mathcal{G}\overset {\exists !}{\to }\mathsf{Core}\webleft (\mathcal{C}\webright )$ making the diagram
commute.
