&\xrightarrow{(C(a^{\#}),C(b^{\#}))} R(g\circ a)_*C^{{\bullet}}({D_Y}_{{\square}},\underline{{\mathbb{C}}}_{{D_Y}_{{\square}}})][-1].\end{aligned}$$

If we shift by $-1$ the short exact sequence of the cone over the morphism $(C(a^{\#}),C(b^{\#}))$ we obtain

$$\begin{aligned}
0&\rightarrow R(g\circ a)_*C^{{\bullet}}({D_Y}_{{\square}},\underline{{\mathbb{C}}}_{{D_Y}_{{\square}}})[-1]\rightarrow\text{Cone}^{{\bullet}}[Rb_*C^{{\bullet}}(\tilde{Y}_{{\square}},\underline{{\mathbb{C}}}_{\tilde{Y}_{{\square}}})\oplus Rg_*C^{{\bullet}}({\Sigma_Y}_{{\square}},\underline{{\mathbb{C}}}_{{\Sigma_Y}_{{\square}}})\xrightarrow{(C(a^{\#}),C(b^{\#}))}\\
&\xrightarrow{(C(a^{\#}),C(b^{\#}))} R(g\circ a)_*C^{{\bullet}}({D_Y}_{{\square}},\underline{{\mathbb{C}}}_{{D_Y}_{{\square}}})][-1]\rightarrow Rb_*C^{{\bullet}}(\tilde{Y}_{{\square}},\underline{{\mathbb{C}}}_{\tilde{Y}_{{\square}}})\oplus Rg_*C^{{\bullet}}({\Sigma_Y}_{{\square}},\underline{{\mathbb{C}}}_{{\Sigma_Y}_{{\square}}})\rightarrow 0\end{aligned}$$

so using the isomorphism above we get the short exact sequence of objects in $D_+(Sh(Y))$

$$0\rightarrow R(g\circ a)_*C^{{\bullet}}({D_Y}_{{\square}},\underline{{\mathbb{C}}}_{{D_Y}_{{\square}}})[-1]\rightarrow C^{{\bullet}}(Y_{{\square}},\underline{{\mathbb{C}}}_{Y_{{\square}}})\rightarrow Rb_*C^{{\bullet}}(\tilde{Y}_{{\square}},\underline{{\mathbb{C}}}_{\tilde{Y}_{{\square}}})\oplus Rg_*C^{{\bullet}}({\Sigma_Y}_{{\square}},\underline{{\mathbb{C}}}_{{\Sigma_Y}_{{\square}}})\rightarrow 0.$$

Now, since the $(m-2)$-cubical hyperresolution ${\varepsilon}:Y_{{\square}}\rightarrow Y$ is of cohomological descent $C^{{\bullet}}(Y_{{\square}},\underline{{\mathbb{C}}}_{Y_{{\square}}})$ is acyclic; hence, if we denote by $Y_{{\bullet}}$ the $(m-3)$-semisimplicial space associated to $S_{{\square}}$ we can write the following isomorphism in $D_+(Sh(Y))$:

$$\underline{{\mathbb{C}}}_{Y}\xrightarrow{\simeq} R{\varepsilon}_*\underline{{\mathbb{C}}}_{Y_{{\bullet}}}.$$

Since all elements of the $(m-3)$-semisimplicial variety $Y_{{\bullet}}$ are smooth, in $D_+(Sh(Y_{{\bullet}}))$ we have an isomorphism $\underline{{\mathbb{C}}}_{Y_{{\bullet}}}\xrightarrow{\simeq}\Omega^{{\bullet}}_{Y_{{\bullet}}}$, so we can substitute $R{\varepsilon}_*\underline{{\mathbb{C}}}^{{\bullet}}_{Y_{{\bullet}}}$ with $R{\varepsilon}_*\Omega^{{\bullet}}_{Y_{{\bullet}}}$; the same can of course be done with the other $(m-2)$-cubical hyperresolutions in $H(Y)_{{\square}}$.

In this way we obtain a short exact sequence of objects in $D_+(Sh(Y))$

$$\label{eq:SESforY}
0\rightarrow R(g\circ a)_*DR_{D_Y}^{{\bullet}}[-1]\rightarrow DR_Y^{{\bullet}}\rightarrow Rb_*DR_{\tilde{Y}}^{{\bullet}}\oplus Rg_*DR_{{\Sigma_Y}}^{{\bullet}}\rightarrow 0$$

which yields the long exact sequence of algebraic de Rham cohomology groups

$$\label{eq:LECSY}
\dots\rightarrow H_{DR}^{{\bullet}}(\Sigma_Y)\oplus H_{DR}^{{\bullet}}(\tilde{Y})\rightarrow H_{DR}^{{\bullet}}(D_Y)\rightarrow H_{DR}^{{\bullet}+1}(Y)\rightarrow\cdots.$$

We want to apply a similar argument to $H(X)_{{\square}}$. We rewrite it as

$$H(X)_{{\square}}=
\xymatrixcolsep{1.5cm}\xymatrixrowsep{1.5cm}\xymatrix{
{D_X}_{{\square}} \ar[r]^f \ar[d]^a & \tilde{X}_{{\square}} \ar[d]^a\\
{\Sigma_X}_{{\square}} \ar[r]^g & X_{{\square}}}$$

where each entry is an $(m-2)$-cubical hyperresolution of 