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I know this has been asked several times before. Some answers for this has been answered theoretical and some of the by counter example.The theoretical answers for this question are OK I want to give an easy couterexample why $K(A)$ since all the examples given before are more or less complicated so I came to the following answer:

(AB1) failure in $\mathcal{K(A)}$, without triangulated categories

But still don't understand why the counterexample at first answer works :/ First doubt there is why $fg=0$ and the second doubt is why it claims the last morphism there exist and should be homotopical to $g$ and Aldo why it claims the last morpshism cannnot be homotopical to $g$. Sorry but I've been stuck on this for a while and really want to see a couterexample clearly for this

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  • $\begingroup$ If you just want a counterexample, you can look at one of my answers here. Maybe this is one of the examples you saw before, though. $\endgroup$
    – Takumi Murayama
    Oct 10, 2019 at 23:08
  • $\begingroup$ @TakumiMurayana Yes, I still see complicated to follow your argument there so I was looking for a simpler counterexample $\endgroup$
    – Cos
    Oct 10, 2019 at 23:46

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