Let $Q$ be a finite quiver without oriented cycles and let $k$ be a field. Then $A = kQ$ is a finite-dimensional path algebra. Let $e_1, \dots, e_n$ be the idempotents corresponding to the vertices $1, \dots, n$ of $Q$. Then the modules $I(i) = D(e_iA)$ are precisely the indecomposable injective modules.
Question: Assume $I(i)$ is not projective. Then there is an Auslander-Reiten sequence $0 \to \tau I(i) \to E(i) \to I(i) \to 0$. How can we describe the module $E(i)$?
In the article "Representations of Wild quivers" by Otto Kerner, the author seems to implicitly say in the proof of theorem 3.8 that $E(i) = \bigoplus I(j) \oplus \bigoplus \tau I(j')$, where $j$ and $j'$ together run through all vertices in the quiver which are adjacent to $i$. Is this correct? Do you know a reference for this?