Remark

More generally, the hypotheses of the proposition are satisfied when $(G,H, \varphi , \psi , \mu)$ is a "groupoid of groupoids." This means that G and H are groupoids, $\varphi$ and $\psi$ are groupoid morphisms of H into G, making the quadruplet $(G,H, \varphi , \psi )$ into a "quiver in groupoids"; finally, $\mu$ is a law of composition in this "quiver" given by a groupoid morphism $H\times_GH\rightarrow H$, satisfying a certain number of properties expressing the associativity of the law and the fact that every "arrow" is invertible. The reader will moreover remark that the mappings to the fundamental groupoid studied in this work all fall under this case.