Existence of the bounded above left total derived functor

Compose the supplied quasi-inverse $D^{-}(\mathcal{A})\to K^{-}(\mathrm{Proj}\mathcal{A})$ with $K(F)$ and $Q_{\mathcal{B}}$. These are functors, so this gives $LF$ on all arrows, including identities and zero complexes. The homotopy equality $p_Y\tilde{u}=u{p}_X$ for an ordinary map makes $QF(p_X)$ a natural augmentation.

Given $(G,\gamma )$, for a projective complex $P$ put ${\mu }_{QP}={ϵ}_P^{-1}{\gamma }_P$; the augmentation here is invertible by replacement comparison with $P\to P$. For general $X$ define ${\mu }_{QX}=LF(Q{p}_X){\mu }_{Q{P}_X}G(Q{p}_X{)}^{-1}$. This is meaningful since both functors take $Q{p}_X$ to isomorphisms. For a derived arrow, lift its conjugate between projective models; naturality of $\gamma$ on that ordinary homotopy-class map gives naturality of $\mu$ after conjugation.

Naturality of $ϵ$ along $p_X$ and of $\gamma$ gives ${ϵ}_X{\mu }_{QX}={\gamma }_X$. Conversely this equation determines $\mu$ on projective complexes because ${ϵ}_P$ is invertible, and naturality along $Q{p}_X$ forces the formula at every object. Thus the universal comparison exists uniquely. All uses of $F$ required only additivity and preservation of homotopies.