Proposition 6

Let C be a quiver, let G be a groupoid, and let $\varphi$ be a quiver morphism from C into G. There exists a unique groupoid morphism $\varphi '$ from Grp(C) into G such that $\varphi '\circ \theta_C=\varphi$.

Let $\psi$ be the quiver morphism from $\widetilde{C}$ into G such that $\psi (a) =\varphi (a)$ for every vertex $a$ of C and $\psi (f, \varepsilon ) =\varphi (f)^{\varepsilon}$ for every arrow $f$ of C and every $\varepsilon \in \{-1,1\}$. One has $\psi \circ i=\varphi$. By Proposition 5, p. 173, there exists a morphism $\varphi '$ from Grp(C) $=\varpi_C$ into G such that $\varphi '\circ j=\psi$. One then has $\varphi '\circ \theta_C=\varphi '\circ j\circ i=\psi \circ i\widetilde{=}\varphi$. As in the proof of Prop. 5, the uniqueness of $\varphi '$ results from the fact that $\theta_C(C)$ generates the groupoid Grp(C).

Under the hypotheses of Prop. 6, one sometimes says that $\varphi '$ is the groupoid morphism from Grp(C) into G extending the quiver morphism $\varphi$.