The total Cartan-Eilenberg complex computes the derived composite

Apply F2 to $G$ and the complex $F(I^{\bullet })$. Its first page is $E_1^{q,p}=R^{p}G(F(I^q))$. F1 makes this zero for $p>0$; for $p=0$ left exactness identifies it with $GF(I^q)$ and the $d_1$ is its cochain differential. Thus the second page has only the row $p=0$, equal to $H^q(GF(I^{\bullet }))$.

Every differential from page two onward has zero source or target off that row, so the page is stationary. In target degree $n$ its finite filtration has only one potentially nonzero quotient, at $(n,0)$; all earlier successive quotients vanish and the last filtration term is zero. Therefore its bottom edge $H^n(GF(I^{\bullet }))\to H^n(\mathrm{Tot}(GJ))$ is an isomorphism. F3 identifies this exact edge with the augmentation map, proving that augmentation is a quasi-isomorphism.

F4 identifies the source cohomology with $R_{\mathsf{I}}^n(GF)(A)$, and nowhere is $F(I^{\bullet })$ treated as a resolution of $F(A)$. Under DC, or when the relevant change-of-resolution comparisons and homotopies are supplied, this relative identification is independent of $\mathsf{I}$ and natural, so the subscript may then be suppressed. At $n=0$ it agrees with the left-exact augmentation kernel identification, and zero terms or zero complexes satisfy the same one-row argument.