{\overline{gh}}\ {\overline{f}} \ar[d]^{\chi_{gh,f}}\\
{\overline{g}}\ {\overline{hf}} \ar[rr]_{\chi_{g,hf}}&& {\overline{ghf}} {\ar @{} [llu] |{\Downarrow\omega_{g,h,f}}},
}$$

such that the next diagrams are equal for all $g,h,f,k \in G$,

$$\label{omega-axiom} 
\xymatrixrowsep{0.2in}
\xymatrixcolsep{0.2in}
\xymatrix{
  &  & {\overline{gh}}\ {\overline{f}}\ {\overline{f}}\ar[rrrr]^{\chi_{gh,f}\otimes 1_{{\overline{k}}}} {\ar @{} [ddrr] |{\Downarrow\omega_{g,h,f}\otimes 1_{{\overline{k}}}}} &  &  &  & {\overline{ghf}}\ {\overline{k}}\ar[rrdd]^{\chi_{ghf,k}} &  &  \\ 
  &  &  &  &  &  &  &  &  \\ 
 {\overline{g}}\ {\overline{h}}\ {\overline{f}}\ {\overline{k}}\ar[rruu]^{\chi_{g,h}\otimes 1_{{\overline{f}}}\otimes 1_{{\overline{k}}}}  \ar[rrdd]_{1_{{\overline{g}}}\otimes 1_{{\overline{h}}}\otimes \chi_{f,k}}\ar[rrrr]^{1_{{\overline{g}}}\otimes \chi_{h,f}\otimes 1_{{\overline{k}}}} &  &  &  & {\overline{g}}\ {\overline{hf}}\ {\overline{k}}\ar[rruu]^{\chi_{g,hf}\otimes 1_{{\overline{k}}}}\ar[rrdd]^{1_{{\overline{g}}}\otimes \chi_{hf,k}} &  &  &  & {\overline{ghfk}} {\ar @{} [llll] |{\Downarrow \omega_{g,hf,k}}}\\ 
  &  &  &  &  &  &  &  &  \\ 
  &  &  {\overline{g}} \ {\overline{h}}\ {\overline{fk}} {\ar @{} [rrruu] |{\Downarrow 1_{{\overline{g}}}\otimes \omega_{h,f,k}}} \ar[rrrr]_{1_{{\overline{g}}}\otimes \chi_{h,fk}} &  &  &  & {\overline{g}}\ {\overline{hfk}}\ar[rruu]_{\chi_{g,hfk}} &  & 
}$$

$$\xymatrixrowsep{0.2in}
\xymatrixcolsep{0.2in}
\xymatrix{
  &  & {\overline{gh}}\ {\overline{f}}\ {\overline{k}} \ar[ddrr]^{1_{{\overline{gh}}}\otimes\chi_{f,k}} \ar[rrrr]^{\chi_{gh,f}\otimes 1_{{\overline{k}}}}  &  &  &  & {\overline{ghf}}\ {\overline{k}}\ar[rrdd]^{\chi_{ghf,k}} {\ar @{} [lldd] |{\Downarrow \omega_{gh,f,k}}} &  &  \\ 
  &  &  &  &  &  &  &  &  \\ 
 {\overline{g}}\ {\overline{h}}\ {\overline{f}}\ {\overline{k}} \ar[rruu]^{\chi_{g,h}\otimes 1_{{\overline{f}}}\otimes 1_{{\overline{k}}}}   \ar[rrdd]_{1_{{\overline{g}}}\otimes 1_{{\overline{h}}}\otimes \chi_{f,k}}   {\ar @{} [rrrr] |{\Downarrow c_{\chi_{g,h},\chi_{f,k}}}} &  &  &  & {\overline{gh}}\ {\overline{fk}}  \ar[rrrr]_{\chi_{gh,fk}} &  &  &  & {\overline{ghfk}} \\ 
  &  &  &  &  &  &  &  &  \\ 
  &  &  {\overline{g}} \ {\overline{h}}\ {\overline{fk}} \ar[uurr]_{\chi_{g,h}\otimes 1_{{\overline{fk}}}}  \ar[rrrr]_{1_{{\overline{g}}}\otimes \chi_{h,fk}} 