Proposition 4

Let $\mu: H\times_GH\rightarrow H$ be a quiver morphism such that $\varphi \circ \mu=\varphi \circ$ pr$_1$ and $\psi \circ \mu=\psi \circ$ pr$_2$. Suppose that, for every pair $(a, b)$ of vertices of H such that $\varphi (a) =\varphi (b)$, there exists a vertex $c$ of H such that $\varphi (c) =\psi (b)$ and $a=\mu(b, c)$.

Let $A_1$ be a set of vertices of Ker($\varphi , \psi$ ) meeting each of its connected components and let $Z_1$ be the set of elements of $\Omega (\widetilde{\Gamma})$ of the form $((a,1))$, where $a\in A_1$. Let also $A_2$ be a set of vertices of $H\times_GH$ meeting each connected component of $H\times_GH$ and let $Z_2$ be the set of triples of the form $((a,1),(b,1),(\mu(a, b),-1))$, as $(a, b)$ ranges over $A_2$.

Then $\Omega (\widetilde{\Gamma})$ is the smallest distinguished subset of $\Omega (\widetilde{\Gamma})$ containing $Z_1\cup Z_2$. In particular, the conjugacy classes $c(\mathbf{z})$ in Coh($\varphi , \psi$ )$_{a_0}$, where $\mathbf{z}$ ranges over $Z_1\cup Z_2$, generate the kernel of the canonical homomorphism $\beta_a$ of Coh($\varphi , \psi$ )$_{a_0}$ into Coeg($\varphi , \psi$ )$_{\beta(a_0)}$.

Let $Z'_1,Z'_2$ and $Z'$ denote the smallest distinguished subsets of $\Omega (\widetilde{\Gamma})$ containing respectively $Z_1,Z_2$ and $Z_1\cup Z_2($II, p. 197, definition 1). It is required to prove that $Z'$ is equal to $\Omega (\widetilde{\Gamma})$; by II, p. 199, Corollary 1 of Proposition 1 of II, p. 197, the last assertion of the proposition will follow.