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The filtered chain complexes of the lower left, upper left and upper right are
depicted in Figure 1 where $i_{k}$ and $\pi_{k}$ are inclusions and
projections into or off of the $k$-component. We have used the convention of
Notation 2.1.8 grouping subscript and superscript terms together to help keep
track of the components from the tensor products.

We make a general notational remark about how to interpret Figures 1, 2, 3, 8
and 9.

######  Notation A.2.1.

Since we are working with tensor products of filtered chain complexes we are
displaying these diagrams so that the constituent $R$-modules summands in a
fixed cohomological degrees can be read off along a diagonal and similarly for
filtration degrees along the horizontal direction. We label differentials
which for the most part are either a sign times the identity, inclusion of a
summand or projection from a summand, e.g. in Figure 2 the differential of $M$
is given by $d(M)=(-1)^{n}N+O$.

Unlabelled arrows in these diagrams denote an identity matrix and should be
interpreted via adjacency, e.g. in Figure 3 there is an arrow from the sum of
$R\left\\{\underline{C}\right\\}$ and $R\left\\{\underline{E}\right\\}$ to the
sum of
$R\left\\{\underline{F}\right\\}$,$R\left\\{\underline{I}\right\\}$,$R\left\\{\underline{H}\right\\}$
and $R\left\\{\underline{K}\right\\}$. Along the arrow
$R\left\\{\underline{C}\right\\}$ is adjacent to
$R\left\\{\underline{F}\right\\}$ and $R\left\\{\underline{E}\right\\}$ is
adjacent to $R\left\\{\underline{I}\right\\}$ so this arrow encodes a
differential sending $\underline{C}\mapsto\underline{F}$ and
$\underline{E}\mapsto\underline{I}$.

Other more complicated names are given explicit names and described within
this appendix.

The boxes demarcating a copy of a filtered graded $R$-module in Figure 1
illustrate the maps of the pushout Diagram 5: a filtered graded $R$-module of
the component corresponding to
$\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)$ demarcated by a dashed
box is mapped identically or diagonally into the boxes of the same dashed type
in the other two diagrams.

We provide names for the generators of these filtered graded $R$-modules in
Figure 2 where the filtration and cohomological indexing has also been
dropped. In this notation the map
$k_{1}\colon\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)\rightarrow\mathcal{B}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)$
of Diagram 5 is determined on generators by:

(6) $\displaystyle k_{1}(I)=B+C$ $\displaystyle k_{1}(J)=E$ $\displaystyle
k_{1}(K)=F+G$ $\displaystyle k_{1}(L)=H\,,$

and the map
$k_{2}\colon\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)\rightarrow\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{B}_{r+1}(q,m)$
of Equation 5 is determined on generators by:

(7) $\displaystyle k_{2}(I)=N+P$ $\displaystyle k_{2}(J)=Q+T$ $\displaystyle
k_{2}(K)=S$ $\displaystyle k_{2}(L)=U\,.$

One can write down similar equations for the differentials of these
generators, e.g. $d(M)=O+(-1)^{n}N$ that can be read off from Figure 2.

We now need to compute the pushout $\mathrm{dom}(i\boxtimes j)$. For this we
must use the description of colimits given by Lemma 2.1.5, i.e. to compute a
colimit of a diagram $X\colon\mathcal{J}\rightarrow f\mathcal{C}$ we view $X$
as an object of $\mathcal{C}^{\mathbb{Z}_{\infty}}$ via an inclusion functor,
compute the colimit in this category instead and then apply the functor $\rho$
to yield an object of $f\mathcal{C}$.

Diagrams for the filtered chain complexes
$\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)$,
$\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{B}_{r+1}(q,m)$ and
$\mathcal{B}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)$ realised as
$\mathbb{Z}_{\infty}$-indexed diagrams are depicted in Figures 4, 5 and 6
respectively. The colimit in $\mathbb{Z}_{\infty}$-indexed chains can then be
computed pointwise from these three diagrams along with the data of the maps
of Equations 6 and 7. The result is shown in Figure 7 where some
identifications have taken place, so that for example we now write
$\overline{N}$ for the class of the image of $N$ in the pushout. The equation
$\overline{N}+\overline{P}=\overline{B}+\overline{C}$ is in an example of an
identification that takes place in the pushout. The pushout can now be
identified in $f\mathcal{C}$ by applying the functor $\rho$ to Figure 7. This
yields the filtered chain complex given by the upper half of Figure 9 where
the the filtered graded $R$-modules correspond to those of Figure 10.

The notation $R_{(\ast)+(\ast)}^{(\ast)+(\ast)}$ of Notation 2.1.8 is somewhat
meaningless after these identifications have taken place however we maintain
it for clarity of the diagrams.

The differentials labelled in Figure 10 can also be read off from Figure 7.
Those labelled $\alpha,\beta,\gamma$ and $\epsilon$ are given by:

(8) $\displaystyle\alpha(\overline{A})$ $\displaystyle=\overline{B}$
$\displaystyle\beta(\overline{M})$
$\displaystyle=(-1)^{n}\overline{B}+(-1)^{n}\overline{C}+(-1)^{n+1}\overline{P}$
$\displaystyle\gamma(\overline{B})$ $\displaystyle=0$
$\displaystyle\gamma(\overline{C})$ $\displaystyle=\overline{Q}+\overline{T}$
$\displaystyle\gamma(\overline{P})$ $\displaystyle=\overline{T}$
$\displaystyle\epsilon(\overline{B})$ $\displaystyle=(-1)^{n}\overline{F}$
$\displaystyle\epsilon(\overline{C})$ $\displaystyle=(-1)^{n}\overline{G}$
$\displaystyle\epsilon(\overline{P})$
$\displaystyle=(-1)^{n}\overline{F}+(-1)^{n}\overline{G}$

We next turn our attention to the map from this pushout into
$\mathcal{B}_{r+1}(p,n)\otimes\mathcal{B}_{r+1}(q,m)$. This codomain
$\mathcal{B}_{r+1}(p,n)\otimes\mathcal{B}_{r+1}(q,m)$ is given in Figure 8 and
as before we give names to the filtered graded $R$-modules in Figure 3. The
map $i\boxtimes j$ is induced from the map on the generators of Figure 2 with
values in the generators of Figure 3 given by:

(9) $\displaystyle A$ $\displaystyle\mapsto\underline{C}+\underline{E}$
$\displaystyle B$ $\displaystyle\mapsto\underline{F}+\underline{I}$
$\displaystyle C$ $\displaystyle\mapsto\underline{H}+\underline{K}$
$\displaystyle D$ $\displaystyle\mapsto\underline{J}$ $\displaystyle E$
$\displaystyle\mapsto\underline{L}+\underline{N}$ $\displaystyle F$
$\displaystyle\mapsto\underline{M}$ $\displaystyle G$
$\displaystyle\mapsto\underline{O}$ $\displaystyle H$
$\displaystyle\mapsto\underline{P}$ $\displaystyle M$
$\displaystyle\mapsto\underline{B}+\underline{D}$ $\displaystyle N$
$\displaystyle\mapsto\underline{F}+\underline{H}$ $\displaystyle O$
$\displaystyle\mapsto\underline{G}$ $\displaystyle P$
$\displaystyle\mapsto\underline{I}+\underline{K}$ $\displaystyle Q$
$\displaystyle\mapsto\underline{L}$ $\displaystyle S$
$\displaystyle\mapsto\underline{M}+\underline{O}$ $\displaystyle T$
$\displaystyle\mapsto\underline{N}$ $\displaystyle U$
$\displaystyle\mapsto\underline{P}$

which can be checked to be compatible with the maps from the generators
$I,J,K$ and $K$ into either component of the pushout. These maps then induce
maps from the pushout in $\mathbb{Z}_{\infty}$-indexed chains to
$\mathcal{B}_{r+1}(p,n)\otimes\mathcal{B}_{r+1}(q,m)$ viewed as a
$\mathbb{Z}_{\infty}$-indexed chain complex.

This completes the description of the pushout-product $i\boxtimes j$ and the
proof of Proposition 5.3.4.

(10) $\displaystyle\alpha_{1}$ $\displaystyle=\begin{pmatrix}1\\\
(-1)^{n}\end{pmatrix}$ $\displaystyle\alpha_{2}$
$\displaystyle=\begin{pmatrix}(-1)^{n+1}&1\end{pmatrix}$
$\displaystyle\beta_{1}$ $\displaystyle=\begin{pmatrix}(-1)^{n}\\\ 1\\\
0\end{pmatrix}$ $\displaystyle\beta_{2}$
$\displaystyle=\begin{pmatrix}1&(-1)^{n+1}&0\\\ 0&0&(1)^{n}\\\
0&0&1\end{pmatrix}$ $\displaystyle\beta_{3}$
$\displaystyle=\begin{pmatrix}(-1)^{n+1}&0\\\ 0&(-1)^{n}\\\ 0&1\end{pmatrix}$
$\displaystyle\beta_{4}$
$\displaystyle=\begin{pmatrix}0&1&(-1)^{n+1}\end{pmatrix}$
$\displaystyle\beta_{5}$ $\displaystyle=(-1)^{n+1}$ $\displaystyle\gamma_{1}$
$\displaystyle=\begin{pmatrix}1\\\ 0\\\ (-1)^{n-1}\end{pmatrix}$
$\displaystyle\gamma_{2}$ $\displaystyle=\begin{pmatrix}0&1&0\\\
(-1)^{n}&0&1\\\ 0&(-1)^{n}&0\end{pmatrix}$ $\displaystyle\gamma_{3}$
$\displaystyle=\begin{pmatrix}1&0\\\ 0&1\\\ (-1)^{n}&0\end{pmatrix}$
$\displaystyle\gamma_{4}$
$\displaystyle=\begin{pmatrix}(-1)^{n+1}&0&1\end{pmatrix}$
$\displaystyle\gamma_{5}$ $\displaystyle=1$

${R_{(p+r)+(q)}^{(n-1)+(m)}}$${R_{(p+r)+(q-r-1)}^{(n-1)+(m+1)}}$${R_{(p)+(q)}^{(n)+(m)}}$${R_{(p)+(q-r-1)}^{(n)+(m+1)}}$${R_{(p-1)+(q)}^{(n)+(m)}}$${R_{(p-1)+(q-r-1)}^{(n)+(m+1)}}$${R_{(p-r-1)+(q)}^{(n+1)+(m)}}$${R_{(p-r-1)+(q-r-1)}^{(n+1)+(m+1)}}$${R_{(p)+(q)}^{(n)+(m)}}$${R_{(p)+(q-r-1)}^{(n)+(m+1)}}$${R_{(p-r-1)+(q)}^{(n+1)+(m)}}$${R_{(p-r-1)+(q-r-1)}^{(n+1)+(m+1)}}$${R_{(p)+(q+r)}^{(n)+(m-1)}}$${R_{(p)+(q)}^{(n)+(m)}\oplus\
R_{(p)+(q-1)}^{(n)+(m)}}$${R_{(p)+(q-r-1)}^{(n)+(m+1)}}$${R_{(p-r-1)+(q+r)}^{(n+1)+(m-1)}}$${R_{(p-r-1)+(q)}^{(n+1)+(m)}\oplus\
R_{(p-r-1)+(q-1)}^{(n+1)+(m)}}$${R_{(p-r-1)+(q-r-1)}^{(n+1)+(m+1)}}$$\scriptstyle{(-1)^{n-1}}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{1}}$${\oplus}$${\oplus}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(-1)^{n+1}}$$\scriptstyle{(-1)^{n}\mathrm{id}}$$\scriptstyle{(-1)^{n}}$$\scriptstyle{(-1)^{n+1}}$$\scriptstyle{(-1)^{n}i_{1}}$$\scriptstyle{(-1)^{n}\pi_{2}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{(-1)^{n+1}i_{1}}$$\scriptstyle{(-1)^{n+1}\pi_{2}}$
Figure 1. Components of the pushout:
$\mathcal{B}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)$,
$\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)$ and
$\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{B}_{r+1}(q,m)$. The central diagram
maps via $k_{1}$ into the top diagram and via $k_{2}$ into the bottom.
${R\left\\{A\right\\}}$${R\left\\{D\right\\}}$${\begin{matrix}R\left\\{B\right\\}\\\
\oplus\\\
R\left\\{C\right\\}\end{matrix}}$${\begin{matrix}R\left\\{F\right\\}\\\
\oplus\\\
R\left\\{G\right\\}\end{matrix}}$${R\left\\{E\right\\}}$${R\left\\{H\right\\}}$${R\left\\{I\right\\}}$${R\left\\{K\right\\}}$${R\left\\{J\right\\}}$${R\left\\{L\right\\}}$${R\left\\{M\right\\}}$${R\left\\{N\right\\}\oplus
R\left\\{P\right\\}}$${R\left\\{S\right\\}}$${R\left\\{O\right\\}}$${R\left\\{Q\right\\}\oplus
R\left\\{T\right\\}}$${R\left\\{U\right\\}}$$\scriptstyle{(-1)^{n-1}}$$\scriptstyle{(-1)^{n}\mathrm{id}}$$\scriptstyle{(-1)^{n+1}}$$\scriptstyle{(-1)^{n}}$$\scriptstyle{(-1)^{n+1}}$$\scriptstyle{(-1)^{n}i_{1}}$$\scriptstyle{(-1)^{n}\pi_{2}}$$\scriptstyle{(-1)^{n+1}i_{1}}$$\scriptstyle{(-1)^{n+1}\pi_{2}}$
Figure 2. Named components of the pushout corresponding to Figure 1
${R\left\\{\underline{A}\right\\}}$${\begin{matrix}R\left\\{\underline{C}\right\\}\amsamp\oplus\amsamp
R\left\\{\underline{E}\right\\}\end{matrix}}$${R\left\\{\underline{J}\right\\}}$${\begin{matrix}R\left\\{\underline{B}\right\\}\\\
\oplus\\\
R\left\\{\underline{D}\right\\}\end{matrix}}$${\begin{matrix}R\left\\{\underline{F}\right\\}\amsamp\oplus\amsamp
R\left\\{\underline{I}\right\\}\\\ \oplus\amsamp\amsamp\oplus\\\
R\left\\{\underline{H}\right\\}\amsamp\oplus\amsamp
R\left\\{\underline{K}\right\\}\end{matrix}}$${\begin{matrix}R\left\\{\underline{M}\right\\}\\\
\oplus\\\
R\left\\{\underline{O}\right\\}\end{matrix}}$${R\left\\{\underline{G}\right\\}}$${\begin{matrix}R\left\\{\underline{L}\right\\}\amsamp\oplus\amsamp
R\left\\{\underline{N}\right\\}\end{matrix}}$${R\left\\{\underline{P}\right\\}}$$\scriptstyle{(-1)^{n-1}i_{1}}$$\scriptstyle{i_{1}}$$\scriptstyle{(-1)^{n-1}\pi_{2}}$$\scriptstyle{i_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(-1)^{n+1}i_{1}}$$\scriptstyle{(-1)^{n+1}\pi_{2}}$
Figure 3. Named codomain of the pushout-product $i\boxtimes j$
n+1+m+1n+1+mn+mp-r-1+q-r-2p-r-1+q-r-1$\vdots$p+q-r-1$\vdots$p+q$\vdots$0$\vdots$$\vdots$$I$$\vdots$0$\vdots$$\vdots$$J\oplus
K$$\vdots$$J\oplus K$$\vdots$0$L$$\vdots$$L$$\vdots$$\alpha_{1}$$\alpha_{2}$
Figure 4. Representation in $\mathbb{Z}_{\infty}$-chains of
$\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)$ with differentials
described in Equation 10
n+1+m+1n+1+mn+mn-1+mp-r-1+q-r-2p-r-1+q-r-1$\vdots$p-r-1+q-1p+q-r-1$\vdots$p+q-1p+q$\vdots$p+q+r$\vdots$0$\vdots$$\vdots$$M$$\vdots$0$\vdots$$\vdots$$\phantom{N\oplus\
}O\oplus P$$N\oplus O\oplus P$$\vdots$$N\oplus O\oplus
P$$\vdots$0$\vdots$$\vdots$$\phantom{Q\oplus S\oplus\ }T$$Q\oplus S\oplus
T$$\vdots$$Q\oplus S\oplus T$$Q\oplus S\oplus
T$$\vdots$$\vdots$0$U$$\vdots$$U$$U$$\vdots$$\vdots$$\beta_{1}$$\beta_{2}$$\beta_{3}$$\beta_{4}$$\beta_{5}$
Figure 5. Representation in $\mathbb{Z}_{\infty}$-chains of
$\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{B}_{r+1}(q,m)$ with differentials
described in Equation 10
n+1+m+1n+1+mn+mn-1+mp-r-1+q-r-2p-r-1+q-r-1$\vdots$p-1+q-r-1p+q-r-1$\vdots$p+q-1p+q$\vdots$p+r+q$\vdots$0$\vdots$$\vdots$$A$$\vdots$0$\vdots$$\vdots$$\phantom{B\oplus\
}C\oplus D$$B\oplus C\oplus D$$\vdots$$B\oplus C\oplus
D$$\vdots$0$\vdots$$\vdots$$\phantom{E\oplus F\oplus\ }G$$E\oplus F\oplus
G$$\vdots$$E\oplus F\oplus G$$E\oplus F\oplus
G$$\vdots$$\vdots$0$H$$\vdots$$H$$H$$\vdots$$\vdots$$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$$\gamma_{4}$$\gamma_{5}$
Figure 6. Representation in $\mathbb{Z}_{\infty}$-chains of
$\mathcal{B}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)$ with differentials
described in Equation 10
n+1+m+1n+1+mn+mn-1+mp-r-1+q-r-2p-r-1+q-r-1$\vdots$p-1+q-r-1p+q-r-1$\vdots$p+q-1p+q$\vdots$p+r+q$\vdots$0$\vdots$$\vdots$$\overline{A}\oplus\overline{M}$$\vdots$0$\vdots$$\vdots$$\phantom{\overline{B}\oplus\
}\overline{C}\oplus\overline{D}\oplus\overline{O}\oplus\overline{P}$$\overline{B}\oplus\overline{C}\oplus\overline{D}\oplus\overline{O}\oplus\overline{P}$$\overline{B}\oplus\overline{C}\oplus\overline{D}\oplus\overline{O}\oplus\overline{P}$$\vdots$0$\vdots$$\vdots$$\phantom{\overline{F}\oplus\overline{Q}}\overline{G}\oplus\overline{T}$$\overline{F}\oplus\overline{Q}\oplus\overline{G}\oplus\overline{T}$$\vdots$$\overline{F}\oplus\overline{Q}\oplus\overline{G}\oplus\overline{T}$$\overline{F}\oplus\overline{Q}\oplus\overline{G}\oplus\overline{T}$$\vdots$0$\overline{L}$$\vdots$$\overline{L}$$\overline{L}$$\vdots$$\vdots$${\scriptstyle\begin{pmatrix}1&(-1)^{n}\\\
0&(-1)^{n}\\\ (-1)^{n-1}&0\\\ 0&1\\\
0&(-1)^{n+1}\end{pmatrix}}$${\scriptstyle\begin{pmatrix}(-1)^{n}&0&1&0&(-1)^{n}\\\
0&1&0&(-1)^{n+1}&0\\\ 0&(-1)^{n}&0&0&(-1)^{n}\\\
0&1&0&0&1\end{pmatrix}}$${\scriptstyle\begin{pmatrix}0&1&0&(-1)^{n}\\\
1&0&(-1)^{n+1}&0\\\ (-1)^{n}&0&0&(-1)^{n}\\\
1&0&0&1\end{pmatrix}}$${\scriptstyle\begin{pmatrix}0&0&1&(-1)^{n+1}\end{pmatrix}}$${\scriptstyle\begin{pmatrix}1&(-1)^{n+1}\end{pmatrix}}$
Figure 7. Representation in $\mathbb{Z}_{\infty}$-chains of the pushout
${R_{(p+r)+(q+r)}^{(n-1)+(m-1)}}$${R_{(p+r)+(q)}^{(n-1)+(m)}}$${R_{(p+r)+(q-1)}^{(n-1)+(m)}}$${R_{(p+r)+(q-r-1)}^{(n-1)+(m+1)}}$${R_{(p)+(q+r)}^{(n)+(m-1)}}$${R_{(p)+(q)}^{(n)+(m)}}$${R_{(p)+(q-1)}^{(n)+(m)}}$${R_{(p)+(q-r-1)}^{(n)+(m+1)}}$${R_{(p-1)+(q+r)}^{(n)+(m-1)}}$${R_{(p-1)+(q)}^{(n)+(m)}}$${R_{(p-1)+(q-1)}^{(n)+(m)}}$${R_{(p-1)+(q-r-1)}^{(n)+(m+1)}}$${R_{(p-r-1)+(q+r)}^{(n+1)+(m-1)}}$${R_{(p-r-1)+(q)}^{(n+1)+(m)}}$${R_{(p-r-1)+(q-1)}^{(n+1)+(m)}}$${R_{(p-r-1)+(q-r-1)}^{(n+1)+(m+1)}}$$\scriptstyle{(-1)^{n-1}i_{1}}$$\scriptstyle{i_{1}}$${\oplus}$$\scriptstyle{(-1)^{n-1}\pi_{2}}$$\scriptstyle{i_{1}}$${\oplus}$${\oplus}$${\oplus}$${\oplus}$${\oplus}$$\scriptstyle{\pi_{2}}$${\oplus}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(-1)^{n+1}i_{1}}$${\oplus}$$\scriptstyle{(-1)^{n+1}\pi_{2}}$$\scriptstyle{id}$$\scriptstyle{(-1)^{n}id}$$\scriptstyle{(-1)^{n}id}$$\scriptstyle{id}$
Figure 8. Codomain of the pushout-product $i\boxtimes j$
${R_{(p+r)+(q)}^{n+m-1}}$${R_{(p+r)+(q-r-1)}^{n+m}}$${R_{(p)+(q+r)}^{n+m-1}}$${\begin{matrix}R_{(p)+(q)}^{n+m}\amsamp\oplus\amsamp
R_{(p)+(q-1)}^{n+m}\\\ \oplus\amsamp\amsamp\\\
R_{(p-1)+(q)}^{n+m}\end{matrix}}$${\begin{matrix}R_{(p)+(q-r-1)}^{n+m+1}\\\
\oplus\\\
R_{(p-1)+(q-r-1)}^{n+m+1}\end{matrix}}$${R_{(p-r-1)+(q+r)}^{n+m}}$${\begin{matrix}R_{(p-r-1)+(q)}^{n+m+1}\amsamp\oplus\amsamp
R_{(p-r-1)+(q-1)}^{n+m+1}\end{matrix}}$${R_{(p-r-1)+(q-r-1)}^{n+m+2}}$${R_{(p+r)+(q+r)}^{(n-1)+(m-1)}}$${\begin{matrix}R_{(p+r)+(q)}^{(n-1)+(m)}\amsamp\oplus\amsamp
R_{(p+r)+(q-1)}^{(n-1)+(m)}\end{matrix}}$${R_{(p+r)+(q-r-1)}^{(n-1)+(m+1)}}$${\begin{matrix}R_{(p)+(q+r)}^{(n)+(m-1)}\\\
\oplus\\\
R_{(p-1)+(q+r)}^{(n)+(m-1)}\end{matrix}}$${\begin{matrix}R_{(p)+(q)}^{(n)+(m)}\amsamp\oplus\amsamp
R_{(p)+(q-1)}^{(n)+(m)}\\\ \oplus\amsamp\amsamp\oplus\\\
R_{(p-1)+(q)}^{(n)+(m)}\amsamp\oplus\amsamp
R_{(p-1)+(q-1)}^{(n)+(m)}\end{matrix}}$${\begin{matrix}R_{(p)+(q-r-1)}^{(n)+(m+1)}\\\
\oplus\\\
R_{(p-1)+(q-r-1)}^{(n)+(m+1)}\end{matrix}}$${R_{(p-r-1)+(q+r)}^{(n+1)+(m-1)}}$${\begin{matrix}R_{(p-r-1)+(q)}^{(n+1)+(m)}\amsamp\oplus\amsamp
R_{(p-r-1)+(q-1)}^{(n+1)+(m)}\end{matrix}}$${R_{(p-r-1)+(q-r-1)}^{(n+1)+(m+1)}}$$\scriptstyle{(-1)^{n-1}}$$\scriptstyle{\alpha}$$\scriptstyle{i_{1}}$$\scriptstyle{1}$$\scriptstyle{\beta}$$\scriptstyle{\epsilon}$$\scriptstyle{\gamma}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(-1)^{n+1}i_{1}}$$\scriptstyle{(-1)^{n+1}\pi_{2}}$$\scriptstyle{i\boxtimes
j}$$\scriptstyle{(-1)^{n-1}i_{1}}$$\scriptstyle{i_{1}}$$\scriptstyle{(-1)^{n-1}\pi_{2}}$$\scriptstyle{i_{1}}$$\scriptstyle{(-1)^{n}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(-1)^{n}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(-1)^{n+1}i_{1}}$$\scriptstyle{(-1)^{n+1}\pi_{2}}$
Figure 9. The pushout-product of generating cofibrations
${R\left\\{\overline{A}\right\\}}$${R\left\\{\overline{D}\right\\}}$${R\left\\{\overline{M}\right\\}}$${\begin{matrix}R\left\\{\overline{B}\right\\}\amsamp\oplus\amsamp
R\left\\{\overline{P}\right\\}\\\ \oplus\amsamp\amsamp\\\
R\left\\{\overline{C}\right\\}\amsamp\amsamp\end{matrix}}$${\begin{matrix}R\left\\{\overline{F}\right\\}\\\
\oplus\\\
R\left\\{\overline{G}\right\\}\end{matrix}}$${R\left\\{\overline{O}\right\\}}$${\begin{matrix}R\left\\{\overline{Q}\right\\}\amsamp\oplus\amsamp
R\left\\{\overline{T}\right\\}\end{matrix}}$${R\left\\{\overline{L}\right\\}}$${R\left\\{\underline{A}\right\\}}$${\begin{matrix}R\left\\{\underline{C}\right\\}\amsamp\oplus\amsamp
R\left\\{\underline{E}\right\\}\end{matrix}}$${R\left\\{\underline{J}\right\\}}$${\begin{matrix}R\left\\{\underline{B}\right\\}\\\
\oplus\\\
R\left\\{\underline{D}\right\\}\end{matrix}}$${\begin{matrix}R\left\\{\underline{F}\right\\}\amsamp\oplus\amsamp
R\left\\{\underline{I}\right\\}\\\ \oplus\amsamp\amsamp\oplus\\\
R\left\\{\underline{H}\right\\}\amsamp\oplus\amsamp
R\left\\{\underline{K}\right\\}\end{matrix}}$${\begin{matrix}R\left\\{\underline{M}\right\\}\\\
\oplus\\\
R\left\\{\underline{O}\right\\}\end{matrix}}$${R\left\\{\underline{G}\right\\}}$${\begin{matrix}R\left\\{\underline{L}\right\\}\amsamp\oplus\amsamp
R\left\\{\underline{N}\right\\}\end{matrix}}$${R\left\\{\underline{P}\right\\}}$$\scriptstyle{(-1)^{n-1}}$$\scriptstyle{\alpha}$$\scriptstyle{i_{1}}$$\scriptstyle{\beta}$$\scriptstyle{1}$$\scriptstyle{\epsilon}$$\scriptstyle{\gamma}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(-1)^{n+1}i_{1}}$$\scriptstyle{(-1)^{n+1}\pi_{2}}$$\scriptstyle{i\boxtimes
j}$$\scriptstyle{(-1)^{n-1}i_{1}}$$\scriptstyle{i_{1}}$$\scriptstyle{(-1)^{n-1}\pi_{2}}$$\scriptstyle{i_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(-1)^{n}}$$\scriptstyle{(-1)^{n}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(-1)^{n+1}i_{1}}$$\scriptstyle{(-1)^{n+1}\pi_{2}}$
Figure 10. Named generators of the pushout-product with differentials
described in Equation 8 and the map $i\boxtimes j$ induced by Equation 9

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