is a consequence of the construction we presented in point 3. of Remark \[rmk:cubobj\]: indeed, that construction does not involve the entry $X_{\emptyset}$ of an $(n+1)$-cubical ${\mathcal{C}}$-object, hence all the entries of $X^*_{{\square}}$ can be found in $X_{{\bullet}}$ too (‘bundled together’ by the coproducts); moreover, the augmentation from the entries of $X_{{\bullet}}$ to $X$ are combinations of the augmentations from the entries of $X^*_{{\square}}$ to $X$.

Proof of Theorem 2
==================

Let us write resolution squares for $X$ and $Y$:

$$S(X):=
\xymatrixcolsep{1.5cm}\xymatrixrowsep{1.5cm}\xymatrix{
D_X \ar[r] \ar[d] & \tilde{X} \ar[d]\\
\Sigma_X \ar[r] & X}\hspace{1cm}
S(Y):=
\xymatrixcolsep{1.5cm}\xymatrixrowsep{1.5cm}\xymatrix{
D_Y \ar[r] \ar[d] & \tilde{Y} \ar[d]\\
\Sigma_Y \ar[r] & Y.}$$

Consider the $m$-cubical hyperresolutions $H(Y)_{{\square}}$ of $S(Y)$ and $H(X)_{{\square}}$ of $S(X)$ provided by hypothesis (i). $H(Y)_{{\square}}$ can be rewritten as a $2$-cubical variety of $(m-2)$-cubical varieties which, by Remark \[rmk:iperrescubcub\], are actually $(m-2)$-cubical hyperresolutions of the corresponding entries of $S(Y)$:

$$H(Y)_{{\square}}=
\xymatrixcolsep{1.5cm}\xymatrixrowsep{1.5cm}\xymatrix{
{D_Y}_{{\square}} \ar[r]^f \ar[d]^a & \tilde{Y}_{{\square}} \ar[d]^b\\
{\Sigma_Y}_{{\square}} \ar[r]^g & Y_{{\square}}.}$$

Being a hyperresolution, $H(Y)_{{\square}}$ is in particular of cohomological descent: hence, if $\underline{{\mathbb{C}}}_{H(Y)_{{\square}}}$ denotes the constant sheaf on $H(Y)_{{\square}}$ then $C^{{\bullet}}(H(Y)_{{\square}},\underline{{\mathbb{C}}}_{H(Y)_{{\square}}})$ is acyclic, and the same is true of $C^{{\bullet}}(H(Y)_{{\square}},\underline{{\mathbb{C}}}_{H(Y)_{{\square}}})[2]$; by [@PeSt Corollary 5.28] we deduce the existence of an isomorphism

$$\begin{aligned}
C^{{\bullet}}(Y_{{\square}},\underline{{\mathbb{C}}}_{Y_{{\square}}})\xrightarrow{\simeq}\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^{\#}))}\\
