Corollary

a) With the notation of proposition 4, suppose moreover that the mapping (Som($\varphi$ ), Som($\psi$ )) of Som(H) into Som(G) $\times$ Som(G) is injective and that its image is the graph of an equivalence relation in Som(G). Then, for every vertex $(a, b)$ of $H\times_GH$, there exists a unique vertex $c_{a,b}$ of H such that $\varphi (c_{a,b}) =\varphi (a)$ and $\psi (c_{a,b}) =\psi (b)$.

b) Suppose further that for every arrow $(f, f')$ of $H\times_GH$, there exists an arrow $f''$ of H such that $\varphi (f'') =\varphi (f)$ and $\psi (f'') =\psi (f')$.

Let A be a set of vertices of $H\times_GH$ meeting each of its connected components. Then, the kernel of the morphism $\beta_{a_0}$ is the smallest distinguished subgroup of Coh($\varphi , \psi$ )$_{a_0}$ containing the conjugacy classes $c((a,1),(b,1),(c_{a,b},-1))$, for $(a, b)\in A$.

Let R be the equivalence relation in Som(G) whose graph is the image of the mapping (Som($\varphi$ ), Som($\psi$ )). Let $(a, b)$ be a vertex of $H\times_GH$. Then we have $R\{\varphi (a), \psi (a)\}$ and $R\{\varphi (b), \psi (b)\}$, whence $R\{\varphi (a), \psi (b)\}$ since $\psi (a) =\varphi (b)$. Consequently, there exists a unique vertex $c$ of H such that $\varphi (c) =\varphi (a)$ and $\psi (c) =\psi (b)$; we denote it by $\mu(a, b)$.

For every arrow $(f, f')$ of $H\times_GH$, choose an arrow $f''$ of H such that $\varphi (f'') =\varphi (f)$ and $\psi (f'') =\psi (f)$ and denote it by $\mu(f, f')$.

Thus a quiver morphism $\mu: H\times_GH\rightarrow H$ has been defined such that $\varphi \circ \mu=\varphi \circ$ pr$_1$ and $\psi \circ \mu=\psi \circ$ pr$_2$.

Let $(a, b)$ be a pair of vertices of H such that $\varphi (a) =\varphi (b)$. We have $R\{\varphi (a), \psi (a)\}$ and $R\{\varphi (b), \psi (b)\}$, hence $R\{\psi (b), \psi (a)\}$. Consequently there exists a unique vertex $c$ of H such that $\varphi (c) =\psi (b)$ and $\psi (c) =\psi (a)$. The vertex $\mu(b, c)$ satisfies $\varphi (\mu(b, c)) =\varphi (b) =\varphi (a)$ and $\psi (\mu(b, c)) =$ $\psi (c) =\psi (a)$. Hence $\mu(b, c) =a$ because the mapping (Som($\varphi$ ), Som($\psi$ )) of Som(H) into Som(G) $\times$ Som(G) is injective.

Let $A_1$ be the set of vertices of Ker($\varphi , \psi$ ) and let $Z_1$ be the set of elements of $\Omega (\widetilde{\Gamma})$ of the form $((a,1))$, for $a\in A_1$. Put $A_2= A$ and let $Z_2$ be the set of elements $((a,1),(b,1),(\mu(a, b),-1))$ of $\Omega (\widetilde{\Gamma})$, for $(a, b)\in A_2$. Let $\widetilde{Z}$ (resp. $\widetilde{Z}_1$, resp. $\widetilde{Z}_2$) denote the smallest distinguished subset of $\Omega (\widetilde{\Gamma})$ containing $Z_1\cup Z_2$ (resp. $Z_1$, resp. $Z_2$).

Let $a\in A_1$; one has $\varphi (a) =\psi (a)$ ; let us put $x=\varphi (a)$. The vertex $a$ is the unique vertex of H such that $\varphi (a) =\psi (a) =x$; in particular, one has $\mu(a, a) =a$. The triplet $((a,1),(a,1),(a,-1))$ therefore belong to $Z_2$. It then follows from conditions (i) and (iii) in the definition of a distinguished subset that $((a,1))$ belong to $\widetilde{Z}_2$. Consequently, one has $Z_1\subset \widetilde{Z}_2$, whence $\widetilde{Z} =\widetilde{Z}_2$.

By Proposition 4, $\Omega (\widetilde{\Gamma})$ is the smallest distinguished subset of $\Omega (\widetilde{\Gamma})$ containing $Z_2$ and the kernel of $\beta_{a_0}$ is the smallest distinguished subgroup of Coh($\varphi , \psi$ )$_{a_0}$ containing the conjugacy classes $c(\mathbf{z})$, for $\mathbf{z}\in Z_2$, whence the corollary.