the corresponding entry of $S(X)$. Hypothesis (i) implies in particular that each $(m-2)$-cubical hyperresolution in $H(Y)_{{\square}}$ can be embedded into the corresponding $(m-2)$-cubical hyperresolution in $H(X)_{{\square}}$ as a closed $m$-cubical subvariety; if we pass to semisimplicial objects, we deduce the existence of natural closed immersions

$$H(Y)_{{\bullet}}\hookrightarrow H(X)_{{\bullet}}\hspace{0.5cm}Y_{{\bullet}}\hookrightarrow X_{{\bullet}}\hspace{0.5cm}{\Sigma_Y}_{{\bullet}}\hookrightarrow {\Sigma_X}_{{\bullet}}\hspace{0.5cm}\tilde{Y}_{{\bullet}}\hookrightarrow \tilde{X}_{{\bullet}}\hspace{0.5cm}{D_Y}_{{\bullet}}\hookrightarrow {D_X}_{{\bullet}}$$

and of the corresponding restriction of sections functors, which we shall denote by $\Gamma$.

Now we apply the same argument as before to the complex of sheaves on $H(X)_{{\square}}$ given by $R\Gamma_{H(Y)_{{\square}}}\underline{{\mathbb{C}}}_{H(X)_{{\square}}}$.

We have the following commutative diagram of functors:

$$\xymatrixcolsep{1.5cm}\xymatrixrowsep{1.5cm}\xymatrix{
Sh({D_X}_{{\bullet}}) \ar[r]^{\Gamma_{{D_Y}_{{\bullet}}}} \ar[d]^{{\varepsilon}_*} & Sh({D_X}_{{\bullet}}) \ar[d]^{{\varepsilon}_*}\\
Sh(D_X) \ar[r]^{\Gamma_{D_Y}} & Sh(D_X).}$$ From this we deduce the equality of the total derived functors $R({\varepsilon}_*\circ\Gamma_{{D_Y}_{{\bullet}}})=R(\Gamma_{D_Y}\circ{\varepsilon}_*)$. But pushforwards preserve injective objects, and the same holds for $\Gamma_{{D_Y}_{{\bullet}}}$ because ${D_Y}_{{\bullet}}$ is closed in ${D_X}_{{\bullet}}$; since injective objects are adapted to any functor, we obtain isomorphisms

$$\label{eq:comm1}
R{\varepsilon}_*\circ R\Gamma_{{D_Y}_{{\bullet}}}\simeq R({\varepsilon}_*\circ\Gamma_{{D_Y}_{{\bullet}}})=R(\Gamma_{D_Y}\circ{\varepsilon}_*)\simeq R\Gamma_{D_Y}\circ R{\varepsilon}_*.$$

This commutativity holds for all the restriction of sections functors previously listed.

The $(m-2)$-cubical hyperresolution ${\varepsilon}:X_{{\square}}\rightarrow X$ is of cohomological descent so in $D_+(Sh(X))$ we have an isomorphism $\underline{{\mathbb{C}}}_{X}\xrightarrow{\simeq} R{\varepsilon}_*\underline{{\mathbb{C}}}_{X_{{\bullet}}}$. Moreover all elements of the $(m-3)$-semisimplicial variety $X_{{\bullet}}$ are smooth, so in $D_+(Sh(X_{{\bullet}}))$ we also have an isomorphism $\underline{{\mathbb{C}}}_{X_{{\bullet}}}\xrightarrow{\simeq}\Omega^{{\bullet}}_{X_{{\bullet}}}$. If we combine 