Left total derived functor is independent up to a unique augmentation-compatible natural isomorphism

Write $p_X:P_X\to X$ and $p_X^{\mathrm{\prime }}:P_X^{\mathrm{\prime }}\to X$. The no-roof bijection gives a unique map $c_X:P_X\to P_X^{\mathrm{\prime }}$ in $K$ such that $p_X^{\mathrm{\prime }}{c}_X=p_X$ in $K$. The opposite comparison is inverse by the same uniqueness. This works for zero complexes and identity replacements. Applying additive $F$ preserves these homotopy identities.

For a derived arrow $u:X\to Y$ the two paths between the chosen projective models have the same localized image, hence coincide in $K$ by the no-roof bijection. Thus $QF(c_X)$ is a natural isomorphism commuting with augmentations.

On any projective complex $P$, the augmentation of either construction is invertible: its replacement map is a homotopy equivalence by step 1.1 applied to the identity replacement. Therefore augmentation compatibility forces a comparison at $P$ to be $({ϵ}_P^{\mathrm{\prime }}{)}^{-1}{ϵ}_P$. Naturality along the isomorphism $Q(p_X):Q{P}_X\to QX$ then forces its value at every $X$. This proves precisely the stated uniqueness.