Set $X_n = g_{n!}K_n$ in $D(\mathcal{C}_{total})$. For each $n \geq 1$ we have $$ \Hom(X_n, X_{n - 1}) = \Hom(K_n, g_n^{-1}g_{n - 1!}K_{n - 1}) = \bigoplus\nolimits_{\varphi : [n - 1] \to [n]} \Hom(K_n, f_\varphi^{-1}K_{n - 1}) $$ Thus we get a map $X_n \to X_{n - 1}$ corresponding to the alternating sum of the maps $K_\varphi^{-1} : K_n \to f_\varphi^{-1}K_{n - 1}$ where $\varphi$ runs over $\delta^n_0, \ldots, \delta^n_n$. We can do this because $K_\varphi$ is invertible by assumption (1). Please observe the similarity with the definition of the maps in the proof of Lemma \ref{lemma-simplicial-resolution-Z-site}. We obtain a complex $$ \ldots \to X_2 \to X_1 \to X_0 $$ in $D(\mathcal{C}_{total})$. We omit the computation which shows that the compositions are zero. By Derived Categories, Lemma \ref{derived-lemma-existence-postnikov-system} if we have $$ \Hom(X_i[i - j - 2], X_j) = 0\text{ for }i > j + 2 $$ then we can extend this complex to a Postnikov system. The group is equal to $$ \Hom(K_i[i - j - 2], g_i^{-1}g_{j!}K_j) $$ Again using that $(K_n, K_\varphi)$ is cartesian we see that $g_i^{-1}g_{j!}K_j$ is isomorphic to a finite direct sum of copies of $K_i$. Hence the group vanishes by assumption (2). Let the Postnikov system be given by $Y_0 = X_0$ and distinguished sequences $Y_n \to X_n \to Y_{n - 1} \to Y_n[1]$ for $n \geq 1$. We set $$ K = \text{hocolim} Y_n[n] $$ To finish the proof we have to show that $g_m^{-1}K$ is isomorphic to $K_m$ for all $m$ compatible with the maps $K_\varphi$. Observe that $$ g_m^{-1} K = \text{hocolim} g_m^{-1}Y_n[n] $$ and that $g_m^{-1}Y_n[n]$ is a Postnikov system for $g_m^{-1}X_n$. Consider the isomorphisms $$ g_m^{-1}X_n = \bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}K_n \xrightarrow{\bigoplus K_\varphi} \bigoplus\nolimits_{\varphi : [n] \to [m]} K_m $$ These maps define an isomorphism of complexes $$ \xymatrix{ \ldots \ar[r] & g_m^{-1}X_2 \ar[r] \ar[d] & g_m^{-1}X_1 \ar[r] \ar[d] & g_m^{-1}X_0 \ar[d] \\ \ldots \ar[r] & \bigoplus\nolimits_{\varphi : [2] \to [m]} K_m \ar[r] & \bigoplus\nolimits_{\varphi : [1] \to [m]} K_m \ar[r] & \bigoplus\nolimits_{\varphi : [0] \to [m]} K_m } $$ in $D(\mathcal{C}_m)$ where the arrows in the bottom row are as in the proof of Lemma \ref{lemma-simplicial-resolution-Z-site}. The squares commute by our choice of the arrows of the complex $\ldots \to X_2 \to X_1 \to X_0$; we omit the computation. The bottom row complex has a postnikov tower given by $$ Y'_{m, n} = \left(\bigoplus\nolimits_{\varphi : [n] \to [m]} \mathbf{Z} \to \ldots \to \bigoplus\nolimits_{\varphi : [0] \to [m]} \mathbf{Z}\right)[-n] \otimes^\mathbf{L}_\mathbf{Z} K_m $$ and $\text{hocolim} Y'_{m, n} = K_m$ (please compare with the proof of Lemma \ref{lemma-abelian-postnikov} and Derived Categories, Example \ref{derived-example-key-postnikov}). Applying the second part of Derived Categories, Lemma \ref{derived-lemma-existence-postnikov-system} the vertical maps in the big diagram extend to an isomorphism of Postnikov systems provided we have $$ \Hom(g_m^{-1}X_i[i - j - 1], \bigoplus\nolimits_{\varphi : [j] \to [m]} K_m) = 0\text{ for }i > j + 1 $$ The is true if $\Hom(K_m[i - j - 1], K_m) = 0$ for $i > j + 1$ which holds by assumption (2). Choose an isomorphism given by $\gamma_{m, n} : g_m^{-1}Y_n \to Y'_{m, n}$ of Postnikov systems in $D(\mathcal{C}_m)$. By uniqueness of homotopy colimits, we can find an isomorphism $$ g_m^{-1} K = \text{hocolim} g_m^{-1}Y_n[n] \xrightarrow{\gamma_m} \text{hocolim} Y'_{m, n} = K_m $$ compatible with $\gamma_{m, n}$. \medskip\noindent We still have to prove that the maps $\gamma_m$ fit into commutative diagrams $$ \xymatrix{ f_\varphi^{-1}g_m^{-1}K \ar[d]_{f_\varphi^{-1}\gamma_m} \ar[r]_{K(\varphi)} & g_n^{-1}K \ar[d]^{\gamma_n} \\ f_\varphi^{-1}K_m \ar[r]^{K_\varphi} & K_n } $$ for every $\varphi : [m] \to [n]$. Consider the diagram $$ \xymatrix{ f_\varphi^{-1}(\bigoplus_{\psi : [0] \to [m]} f_\psi^{-1}K_0) \ar@{=}[r] \ar[d]_{f_\varphi^{-1}(\bigoplus K_\psi)} & f_\varphi^{-1}g_m^{-1}X_0 \ar[d] \ar[r]_{X_0(\varphi)} & g_n^{-1}X_0 \ar[d] & \bigoplus_{\chi : [0] \to [n]} f_\chi^{-1}K_0 \ar@{=}[l] \ar[d]^{\bigoplus K_\chi} \\ f_\varphi^{-1}(\bigoplus_{\psi : [0] \to [m]} K_m) \ar@{=}[d] & f_\varphi^{-1}g_m^{-1}K \ar[d]_{f_\varphi^{-1}\gamma_m} \ar[r]_{K(\varphi)} & g_n^{-1}K \ar[d]^{\gamma_n} & \bigoplus_{\chi : [0] \to [n]} K_n \ar@{=}[d] \\ f_\varphi^{-1}Y'_{0, m} \ar[r] & f_\varphi^{-1}K_m \ar[r]^{K_\varphi} & K_n & Y'_{0, n} \ar[l] } $$ The top middle square is commutative as $X_0 \to K$ is a morphism of simplicial objects. The left, resp.\ the right rectangles are commutative as $\gamma_m$, resp.\ $\gamma_n$ is compatible with $\gamma_{0, m}$, resp.\ $\gamma_{0, n}$ which are the arrows $\bigoplus K_\psi$ and $\bigoplus K_\chi$ in the diagram. Going around the outer rectangle of the diagram is commutative as $(K_n, K_\varphi)$ is a simplical system and the map $X_0(\varphi)$ is given by the obvious identifications $f_\varphi^{-1}f_\psi^{-1}K_0 = f_{\varphi \circ \psi}^{-1}K_0$. Note that the arrow $\bigoplus_\psi K_m \to Y'_{0, m} \to K_m$ induces an isomorphism on any of the direct summands (because of our explicit construction of the Postnikov systems $Y'_{i, j}$ above). Hence, if we take a direct summand of the upper left and corner, then this maps isomorphically to $f_\varphi^{-1}g_m^{-1}K$ as $\gamma_m$ is an isomorphism. Working out what the above says, but looking only at this direct summand we conclude the lower middle square commutes as we well. This concludes the proof.