Imported Hom computation into a K-injective complex

Let $\mathcal{A}$ be an abelian category, with the standing size bounds for its derived localization. Use cochain complexes. Write $K(\mathcal{A})$ for their homotopy category and $D(\mathcal{A})$ for its localization at quasi-isomorphisms. Here $H^n(N)=\ker d_N^n\mathrm{/}\mathrm{im}{d}_N^{n-1}$; a complex is acyclic when all these objects vanish, and a quasi-isomorphism induces isomorphisms on all of them. A cochain homotopy has $f-g=dh+hd$, with $h^n:N^n\to I^{n-1}$; this is the chain convention under $N_n=N^{-n}$.

Suppose $I$ is K-injective, meaning ${\mathrm{Hom}}_{K(\mathcal{A})}(M,I)=0$ for every acyclic $M$. Then the canonical map ${\mathrm{Hom}}_{K(\mathcal{A})}(N,I)\to {\mathrm{Hom}}_{D(\mathcal{A})}(N,I)$ is bijective for every complex $N$. For every quasi-isomorphism $s:M\to N$, precomposition gives a bijection ${\mathrm{Hom}}_{K(\mathcal{A})}(N,I)\to {\mathrm{Hom}}_{K(\mathcal{A})}(M,I)$. No boundedness hypothesis is imposed.

Abelian category

Cochain complex in an abelian category

Cohomology object of a cochain complex

A chain homotopy

The homotopy category of chain complexes

Derived category of an abelian category

These two bijections are imported from Stacks, Lemma 13.31.2, conditions (1), (2) and (3). Its proof is cited, rather than reproduced here. This input supplies no K-injective resolution-existence assertion.