as functors from $D_+(Sh({D_Y}_{{\bullet}}))$ to $D_+(Sh(D_X))$.

If we apply to the sides of (\[eq:isosheaves\]) the corresponding $R{\varepsilon}_*$ and use (\[eq:comm2\]) on the left-hand side and (\[eq:comm1\]) on the right-hand side, we obtain an isomorphism between de Rham complexes $j_*DR^{{\bullet}}_{D_Y}\xrightarrow\simeq R\Gamma_{D_Y}DR^{{\bullet}}_{D_X}[2]$ in $D_+(Sh(D_X))$ and so an isomorphism between de Rham cohomology groups. Since we can repeat the previous reasoning for the $m_2$-cubical hyperresolutions of $\Sigma_Y$ and $\Sigma_X$ provided by hypothesis (ii), we obtain isomorphisms

$$\begin{aligned}
\label{eq:morfismibelli}
\begin{split}
&H_{DR}^{{\bullet}}(D_Y)\xrightarrow{\simeq} H_{DR,D_Y}^{{\bullet}+2}(D_X)\\
&H_{DR}^{{\bullet}}(\Sigma_Y)\xrightarrow{\simeq} H_{DR,\Sigma_Y}^{{\bullet}+2c}(\Sigma_X).
\end{split}\end{aligned}$$

These isomorphisms will allow us to relate the long exact sequences (\[eq:LECSY\]) and (\[eq:LECSX\]).

#### $\underline{c=1}$

For $k\geq 1$ we have the following diagram

$$\label{eq:diag1}
\begin{tikzpicture}
  \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
  {
   H_{DR}^{k-1}(\tilde{Y})\oplus H_{DR}^{k-1}(\Sigma_Y) & H_{DR,\tilde{Y}}^{k+1}(\tilde{X})\oplus H_{DR,\Sigma_Y}^{k+1}(\Sigma_X)\\
     H_{DR}^{k-1}(D_Y) & H_{DR,DY}^{k+1}(D_X)\\
     H_{DR}^k(Y) & H_{DR,Y}^{k+2}(X)\\
     H_{DR}^k(\tilde{Y})\oplus H_{DR}^k(\Sigma_Y) & H_{DR,\tilde{Y}}^{k+2}(\tilde{X})\oplus H_{DR,\Sigma_Y}^{k+2}(\Sigma_X)\\
   H_{DR}^k(D_Y) & H_{DR,D_Y}^{k+2}(D_X)\\};
  \path[-stealth]
    (m-1-1) edge node [left] {${\alpha}$} (m-2-1)
        (m-1-1) edge node [above] {$\simeq$} (m-1-2)
        (m-2-1) edge node [left] {$\beta$} (m-3-1)
        (m-2-1) edge node [above] {$\simeq$} (m-2-2)
        (m-3-1) edge node [left] {$\delta$} (m-4-1)
        (m-4-1) edge node [left] {$\sigma$} (m-5-1)
        (m-4-1) edge node [above] {$\simeq$} (m-4-2)
        (m-5-1) edge node [above] {$\simeq$} (m-5-2)
        (m-1-2) edge node [right] {${\alpha}'$} (m-2-2)
    (m-2-2) edge node [right] {$\beta'$} (m-3-2)
        (m-3-2) edge node 