The Stacks project

For anyone that might care: from Lemma 13.31.2 we get that $$\label{1}\tag{1} \operatorname{Hom}_{K(\mathcal{A})}(N^\bullet,I^\bullet)\to\operatorname{Hom}_{K(\mathcal{A})}(I^\bullet,I^\bullet)$$ is bijective. In particular, $I^\bullet\to N^\bullet$ has a left inverse. This alone already implies that $I^\bullet\to I^\bullet$ is a final object of $I^\bullet/\operatorname{Qis}(\mathcal{A})$.

Even though one does not need it in the proof, actually $I^\bullet\to N^\bullet$ is invertible in $K(\mathcal{A})$. Bijectivity of \eqref{1} means that $I^\bullet\to N^\bullet$ has a unique left inverse, and this implies existence of an inverse in a preadditive category. The dual result is proven here.

I don't think the statement in the second sentence of the proof is correct: Let $I^\bullet = 0$ and $N^\bullet$ be acyclic. Then the map $I^\bullet \to N^\bullet$ is a quasi-isomorphism, but, in general, not a homotopy equivalence. (However, as Elías Guisado points out, the existence of a left inverse is sufficient to prove the lemma anyway.)

Sorry, yes, I forgot to answer Elias comment earlier this summer. Thanks for insisting! Now fixed here.

Correcting myself in #9464:

1. “In a preadditive category, a morphism with a unique left inverse is invertible.” This is false; for an example, consider the inclusion $\iota:A[0]\to A[0]\oplus A[1]$ in the category of chain complexes over a preadditive category $\mathcal{A}$, where $A\in\mathcal{A}$ is not zero. Then $\iota$ has a unique left inverse, but is not invertible. What is true is: “in a preadditive category, an endomorphism with a unique left inverse is invertible.”

2. It is also not the case that a quasi-isomorphism $I^\bullet\to N^\bullet$, where $I^\bullet$ is K-injective, needs to be a homotopy equivalence. For an example, take $I^\bullet=0$ and take $N^\bullet$ to be an acyclic but not contractible cochain complex. For instance, $N^\bullet$ could be any non-split short exact sequence of locally finite-free sheaves over a ringed space (here's one). What is true is a quasi-isomorphism with K-injective domain has a unique left homotopy inverse.