The cokernel of an irreducible monomorphism is always the third term of an AR sequence with indecomposable middle term?

Question

I recently studied the structure of the AR quiver of Dynkin type $\mathbb{A}_n/I$, $\mathbb{D}_n/I$ with $I$ any admissable ideal, and found that the cokernel of an irreducible monomorphism is always the third term of an AR sequence with indecomposable middle term, and dually the kernel of an irreducible epimorphism is always the first term of an AR sequence with indecomposable middle term. But I hardly find any idea to prove it. Any suggestions are welcome!

Here irreducible means irreducible morphisms between indecomposable modules.

Answer 1

I believe that this is a result of Brenner and Krause for any finite-dimensional algebra. However, there is a condition that the (co)kernel is not simple.

[1] Brenner, Sheila. On the kernel of an irreducible map. Special issue on linear algebra methods in representation theory. Linear Algebra Appl. 365 (2003), 91–97.

[2] Krause, Henning. The kernel of an irreducible map. Proc. Amer. Math. Soc. 121 (1994), no. 1, 57–66.