0}$ occurrences of $F$. We identify this object with $n$, so that $Fn$ is ${n\pl 1}$. (One can find in [@DP03] a more detailed construction of $\cal S$, which is there called ${\cal L}_c$.) The category $\cal S$ is the disjoint union of the categories $\cal S_A$, whose objects are even, and $\cal S_B$, whose objects are odd.

For $\cal C$ being one of the categories $\cal A$ and $\cal B$ of the penultimate paragraph, and a subscript of one of the categories $\cal S_A$ and $\cal S_B$ of the preceding paragraph, we can prove the following.

We define first by induction the functors $H_{\cal C}$ from $\cal S_C$ to $\cal C$, for $\alpha$ being $\varphi$ or $\gamma$:

=$H_{\cal A}0=0$,=$H_{\cal A}(2n\pl
2)=PH_{\cal B}(2n\pl 1)$,=$H_{\cal B}(2n\pl
1)=UH_{\cal A}2n$,\
$H_{\cal A}\alpha_{2n}=\alpha^{\cal A}_{H_{\cal A}2n}$, $H_{\cal B}\alpha_{2n+1}=\alpha^{\cal B}_{H_{\cal B}(2n+1)}$,\
$H_{\cal C}\mj_n=\mj_{H_{\cal C}n}$,$H_{\cal C}(h_2\cirk
h_1)=H_{\cal C}h_2\cirk H_{\cal C}h_1$,\
$H_{\cal A}Fg=PH_{\cal B}g$,$H_{\cal B}Ff=UH_{\cal A}f$.

Next we define by induction the functors $K_{\cal C}$ from $\cal
C$ to $\cal S_C$:

=$H_{\cal A}0=0$,=$H_{\cal A}(2n\pl
2)=PH_{\cal B}(2n\pl 1)$,=$H_{\cal B}(2n\pl
1)=UH_{\cal A}2n$,

$K_{\cal A}0=0$,$K_{\cal A}PB=K_{\cal B}B\pl 1$,$K_{\cal
B}UA=K_{\cal A}A\pl 1$,\
$K_{\cal C}\alpha^{\cal C}_C=\alpha_{K_{\cal C}C}$,\
$K_{\cal C}\mj_C=\mj_{K_{\cal C}C}$,$K_{\cal C}(h_2\cirk
h_1)=K_{\cal C}h_2\cirk K_{\cal C}h_1$,\
$K_{\cal A}Pg=FK_{\cal B}g$,$K_{\cal B}Uf=FK_{\cal A}f$.

We verify by 