Lemma 22.20.4. In Situation 22.20.2 given any object $x$ of $\mathcal{A}$, and the cone $C(1_ x)$ of the identity morphism $1_ x : x \to x$, the identity morphism on $C(1_ x)$ is homotopic to zero.

**Proof.**
Consider the admissible short exact sequence given by axiom (C).

Then by Lemma 22.20.1, identifying hom-sets under shifting, we have $1_ x=\pi d(s)=-d(\pi )s$ where $s$ is regarded as a morphism in $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{A}}^{-1}(x,C(1_ x))$. Therefore $a=a\pi d(s)=d(s)$ using formula (5) of Lemma 22.20.1, and $b=-d(\pi )sb=-d(\pi )$ by formula (6) of Lemma 22.20.1. Hence

since $s$ is of degree $-1$. $\square$

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