Corollary

The groupoid Coeg($\varphi , \psi$ ) is generated by the image of the morphism $\gamma$.

Let C be the subgroupoid of Coeg($\varphi , \psi$ ) generated by the image of G; let $i$ denote the canonical morphism of C into Coeg($\varphi , \psi$ ) and $\theta : G\rightarrow C$ the morphism such that $i\circ \theta =\gamma$. By proposition 2, there exists a unique groupoid morphism $\overline{\theta}:$ Coeg($\varphi , \psi$ )$\rightarrow C$ such that $\overline{\theta}\circ \gamma =\theta$. Then $\gamma =i\circ \overline{\theta}\circ \gamma$, hence $i\circ \overline{\theta}$ is the identity morphism of Coeg($\varphi , \psi$ ) (loc. cit.). In particular, the mappings Som($i$) and Fl($i$) are surjective, therefore C = Coeg($\varphi , \psi$ ).