I am reading the paper "Cluster automrphisms", here is the link: http://prospero.dmat.usherbrooke.ca/ibrahim/publications/Cluster_Automorphisms.pdf
In the proof of lemma 3.1 I am stuck: For an acyclic quiver $Q$ and a seed $(X,Q)$ of a cluster algebra $\mathcal{A}$, we can get a cluster tilting object $M= \oplus_{x \in X} M_x$ in the cluster category $\mathcal{C}_Q$ correpsonding to $X$. If $EndM \cong kQ$ or $EndM \cong kQ^{op}$, then how to get that the set $\{ M_x \mid x \in X\}$ forms a local slice in $\mathcal{C}_Q$?
The definition of local slice is not short, so you can see section 2.4 of this paper: https://arxiv.org/pdf/1606.05161.pdf
(If $Q$ is of type ADE, then I can get the result by the equivalence: $ind \mathcal{C}_Q/ add \tau M \cong mod EndM$ and the projective component of the AR quiver of $mod End M$ forms a local slice. For the general case, I find it does not work. Thank you for any help.)
