The Stacks project

One could add to the statement:

(4) Any quasi-isomorphism $I^\bullet\to N^\bullet$ is a homotopy equivalence.

Proof of the equivalence. (2)$\Rightarrow$(4). Precomposition by $I^\bullet\to N^\bullet$ induces a bijection $\operatorname{Hom}_{K(\mathcal{A})}(N^\bullet,I^\bullet)\to\operatorname{Hom}_{K(\mathcal{A})}(I^\bullet,I^\bullet)$. This implies that $I^\bullet\to N^\bullet$ has a unique left inverse in $K(\mathcal{A})$. In a preadditive category, this implies invertibility.

(4)$\Rightarrow$(3). A morphism $M^\bullet\to I^\bullet$ in $D(\mathcal{A})$ is of the form $s^{-1}f$, where $f:M^\bullet\to N^\bullet$ and $s:I^\bullet\to N^\bullet$ is a quasi-isomorphism. Hence $\operatorname{Hom}_{K(\mathcal{A})}(M^\bullet,I^\bullet)\to\operatorname{Hom}_{D(\mathcal{A})}(M^\bullet,I^\bullet)$ is onto. It is also injective: if $f:M^\bullet\to I^\bullet$ is a map in $K(\mathcal{A})$ that becomes zero in $D(\mathcal{A})$, then by Homology, Comment #9465 there is a quasi-isomorphism $s:I^\bullet\to N^\bullet$ with $s\circ f=0$ in $K(\mathcal{A})$. Thus $f=0$ in $K(\mathcal{A})$. $\square$

For the interested reader, I wrote the generalization of this result to arbitrary triangulated categories and I merged it with the list of equivalent definitions of K-injective complex from Lipman [L, 2.3.8], including the proof (I claim no originality over any of this). See [GH, Prop. 2.4].

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[GH] —, A Gospel Harmony of Derived Functors https://eliasguisado.wordpress.com/work/

[L] J. Lipman. “Notes on Derived Functors and Grothendieck Duality”. In: Foundations of Grothendieck Duality for Diagrams of Schemes. Lecture Notes in Mathematics. Springer-Verlag, 2009