let $B_j$ be the $n_j$-element set consisting of the canonical basis vectors of $\mathbb E^{n_j}$, and set $X_j=T_{\mathbb S}B_j$. Assume we are given ${{\ensuremath{F_{\,{\mathbb S}}}}}$-coalgebras $(X_1,c_1)$ and $(X_2,c_2)$, and elements $x_j\in X_j$ with $\operatorname{tr}_{c_1}x_1=\operatorname{tr}_{c_2}x_2$.

The extension lemma provides ${{\ensuremath{F_{\,{\mathbb E}}}}}$-coalgebras $(\mathbb E^{n_j},\tilde c_j)$ with $\tilde c_j|_{X_j}=c_j$. Clearly, $\operatorname{tr}_{\tilde c_1}x_1=\operatorname{tr}_{\tilde c_2}x_2$. Using the zig-zag diagram in [[Coalg]{}([$F_{\mathbb E}$]{})]{} and appending inclusion maps, we obtain what we call the *basic diagram*. In this diagram all solid arrows are arrows in E, and all dotted arrows are arrows in . The horizontal dotted arrows denote the inclusion maps, and $\pi_j$ are the restrictions to $Z$ of the canonical projections. $$\xymatrix@C=15pt@R=35pt@M=7pt{
        X_1 \ar@{.>}[r] \ar@{.>}[d]_{c_1} & \mathbb E^{n_1} \ar[d]_{\tilde c_1}
        && Z \ar[ll]_{\pi_1} \ar[rr]^{\pi_2} \ar[d]^d 
            \save[]+<3pt,10pt>*\txt{\begin{rotate}{90}$\subseteq$\end{rotate}}\restore
            \save[]+<2pt,27pt>*\txt{$\mathbb E^{n_1}\!\!\times\mathbb E^{n_2}$}\restore
        && \mathbb E^{n_2} \ar[d]_{\tilde c_2} & X_2 \ar@{.>}[l] \ar@{.>}[d]^{c_2}
        \\
        {{\ensuremath{F_{\,{\mathbb S}}}}}X_1 \ar@{.>}[r] & {{\ensuremath{F_{\,{\mathbb E}}}}}\mathbb E^{n_1} 
        && {{\ensuremath{F_{\,{\mathbb E}}}}}Z \ar[ll]_{{{\ensuremath{F_{\,{\mathbb E}}}}}\pi_1} \ar[rr]^{{{\ensuremath{F_{\,{\mathbb E}}}}}\pi_2}
            \save[]+<-3pt,-10pt>*\txt{\begin{rotate}{-90}$\subseteq$\end{rotate}}\restore
            \save[]+<2pt,-29pt>*\txt{$\mathbb E\!\!\times(\mathbb E^{n_1}\!\!\times\mathbb E^{n_2})^A$}\restore
        && {{\ensuremath{F_{\,{\mathbb E}}}}}\mathbb E^{n_2} & {{\ensuremath{F_{\,{\mathbb S}}}}}X_2 \ar@{.>}[l]
    }$$ Commutativity of this diagram yields $d\big(\pi_j^{-1}(X_j)\big)\subseteq({{\ensuremath{F_{\,{\mathbb E}}}}}\pi_j)^{-1}\big({{\ensuremath{F_{\,{\mathbb S}}}}}X_j)$ for $j=1,2$. Now we observe the following properties of cubic functors.

\[CubProperties\] We have ${{\ensuremath{F_{\,{\mathbb E}}}}}X\cap{{\ensuremath{F_{\,{\mathbb S}}}}}Y={{\ensuremath{F_{\,{\mathbb S}}}}}(X\cap Y)$. Moreover, if $Y_j\subseteq X_j$, then $({{\ensuremath{F_{\,{\mathbb E}}}}}\pi_1)^{-1}({{\ensuremath{F_{\,{\mathbb S}}}}}Y_1)\cap({{\ensuremath{F_{\,{\mathbb E}}}}}\pi_2)^{-1}({{\ensuremath{F_{\,{\mathbb S}}}}}Y_2)
        ={{\ensuremath{F_{\,{\mathbb 