Existence of the bounded below right total derived functor

Compose the supplied injective-model quasi-inverse with $K(F)$ and then $Q_{\mathcal{B}}$. This constructs the functor, and also its factorization before $Q_{\mathcal{B}}$. The identity $\tilde{u}{j}_X=j_{Y}u$ in $K$ proves naturality of ${\eta }_X=QF(j_X)$. The construction includes zero complexes.

For a second system $j_X^{\mathrm{\prime }}:X\to I_X^{\mathrm{\prime }}$, the no-roof bijection supplies a unique homotopy class $c_X:I_X\to I_X^{\mathrm{\prime }}$ with $c_X{j}_X=j_X^{\mathrm{\prime }}$. The reverse comparison is inverse by uniqueness. The same uniqueness on conjugated derived arrows gives naturality after $F$. For an injective complex $I$, $j_I$ is therefore a homotopy equivalence and ${\eta }_I$ is invertible.

Given $(G,\gamma )$ put ${\nu }_{QI}={\gamma }_I{\eta }_I^{-1}$ on injective complexes. On $X$ put ${\nu }_{QX}=G(Q{j}_X{)}^{-1}{\nu }_{Q{I}_X}RF(Q{j}_X)$. Conjugating any derived arrow to its unique injective-model homotopy class proves naturality. Naturality along $j_X$ gives ${\nu }_{QX}{\eta }_X={\gamma }_X$. Any such comparison must have the prescribed value on injectives and then on $X$, so it is unique. Applying this property to two replacement functors proves uniqueness of the isomorphism relative to coaugmentations.

The factorization in step 1.1 takes values in bounded-below complexes because $F$ preserves zero objects. The target localization and then the fully faithful bounded embedding send precisely this complex to its unbounded derived class. Thus the two descriptions agree; neither asserts an unbounded injective replacement theorem.