Proposition 3

The quiver morphism $\alpha$ is a groupoid morphism from G into Coh($\varphi , \psi$ ), and the mapping $h$ is a homotopy joining $\alpha \circ \varphi$ to $\alpha \circ \psi$.

The triple (Coh($\varphi , \psi$ )$, \alpha , h$) has the following universal property: if $G'$ is a groupoid, $\alpha '$ a morphism of groupoids from G into $G'$ and $h':$ Som(H) $\rightarrow$ Fl(G$'$) a homotopy joining $\alpha '\circ \varphi$ to $\alpha '\circ \psi$, there exists a unique morphism of groupoids $\eta :$ Coh($\varphi , \psi$ )$\rightarrow G'$ such that

(3) $\alpha '=\eta \circ \alpha$ and $h'=$ Fl($\eta$ )$\circ h$.

In view of the definition of the groupoid deduced by contraction of arrows, the contraction of the loops (1) implies that $\alpha$ is a morphism of groupoids, and the contraction of the loops (2) that $h$ is a homotopy joining $\alpha \circ \varphi$ to $\alpha \circ \psi$.

Let $G',\alpha ',h'$ be as in the statement. Let $\eta_1$ be the morphism of quivers from $G_1$ into $G'$ such that Som($\eta_1$) is equal to Som($\alpha '$) and such that Fl($\eta_1$) coincides with Fl($\alpha '$) on Fl(G) and with $h'$ on Som(H). There exists a unique morphism of groupoids $\eta_2:$ Grp(G$_1$)$\rightarrow G'$ such that $\eta_1=\eta_2\circ \theta_1$. Since $\alpha '$ is a morphism of groupoids and $h'$ is a homotopy joining $\alpha '\circ \varphi$ to $\alpha '\circ \psi ,\eta_2$ defines by passing to the quotient a morphism of groupoids $\eta$ from Coh($\varphi , \psi$ ) into $G'($II, p. 170, proposition 3). This morphism satisfies relations (3), and it is the only one (II, p. 185, proposition 2).