Injective complexes model the bounded below derived category

Bounded-below injective complexes are K-injective; their Hom groups into each other agree in $K$ and $D$ by the no-roof theorem and in $D^{+}$ by its fully faithful embedding. This proves full faithfulness, including the zero complex.

Enough injectives supplies objectwise replacements under the stated choices. For the simultaneously supplied $j_X$, assign $R(X)=I_X$ and let $R(u)$ be the unique homotopy class whose image is $Q(j_Y)uQ(j_X{)}^{-1}$. Full faithfulness proves functoriality and the quasi-inverse identities. Finite sums, cones and shifts stay in bounded-below injectives; cone triangles therefore give exactness of the equivalence, by lifting first arrows and comparing triangle completions.

For the testing assertion suppose $I^j=0$ below $a$. Fix an acyclic $A$ and an integer $r$. The three terms in degrees $r-1,r,r+1$ of $\underset{}{\mathrm{Hom}}(A,I)$, and their differentials, depend only on terms of $A$ strictly above $c=a-r-3$. Thus replacing $A$ by ${\tau }^{\ge c}A$ leaves these terms and maps unchanged: any factor involving the modified cut has target $I^j=0$. This truncation is bounded below and acyclic. Its Hom cohomology in degree $r$ vanishes by the restricted test, so the original Hom cohomology does too. Since $r$ was arbitrary, $I$ is K-injective. The converse is immediate by restricting the acyclic inputs.