A bounded below complex of injectives is homotopically injective

Let $A$ be acyclic and $f:A\to I$ a chain map. Choose $a$ with $I^n=0$ for $n<a$ and set $h^n:A^n\to I^{n-1}$ equal to zero for $n\le a$. The equation $f^n=d_I{h}^n+h^{n+1}{d}_A$ holds below $a$. In particular it is valid for zero complexes.

If the equation holds below $n$, then $u^n=f^n-d_I^{n-1}{h}^n$ vanishes on $\mathrm{im}{d}_A^{n-1}$ by the chain-map identity. Since $A$ is acyclic this image equals $\ker d_A^n$, so $u^n$ factors through $\mathrm{im}{d}_A^n\hookrightarrow A^{n+1}$. Injectivity of $I^n$ extends it to $h^{n+1}:A^{n+1}\to I^n$. This establishes the equation in degree $n$.

Apply DC to the set of finite partial homotopies with the entire extension relation, or take the supplied successive extensions. The resulting homotopy makes $f$ zero in $K$. Apply the same argument to every shift $I[r]$, which remains bounded below and termwise injective. This is the K-injective condition.