{\overline{gh}}\ {\overline{f}}\ A_k \ar[ddrr]^{1_{{\overline{gh}}}\otimes\theta_{f,k}} \ar[rrrr]^{\chi_{gh,f}^0}  &  &  &  & {\overline{ghf}}\ A_k\ar[rrdd]^{\theta_{ghf,k}} {\ar @{} [lldd] |{\Downarrow \alpha_{gh,f,k}}} &  &  \\ 
  &  &  &  &  &  &  &  &  \\ 
 {\overline{g}}\ {\overline{h}}\ {\overline{f}}\ A_k \ar[rruu]^{\chi_{g,h}^0\otimes 1_{{\overline{f}}}}   \ar[rrdd]_{1_{{\overline{g}}}\otimes 1_{{\overline{h}}}\otimes \theta_{f,k}}   {\ar @{} [rrrr] |{\Downarrow (\chi_{g,h})_{\theta_{f,k}}}} &  &  &  & {\overline{gh}}\ A_{fk}  \ar[rrrr]_{\theta_{gh,fk}} &  &  &  & A_{ghfk} \\ 
  &  &  &  &  &  &  &  &  \\ 
  &  &  {\overline{g}} \ {\overline{h}}\ A_{fk} \ar[uurr]_{\chi_{g,h}^0}  \ar[rrrr]_{1_{{\overline{g}}}\otimes \theta_{h,fk}} &  &  &  & {\overline{g}}\ A_{hfk}\ar[rruu]_{\theta_{g,hfk}} {\ar @{} [lluu] |{\Downarrow\alpha_{g,h,fk}}} &  & 
}$$ holds in ${{\mathcal B}}(F_g (F_h(F_f(A_k)), A_{ghfk})$. If $(A,\theta, \alpha)$ is a 0-cell, the identity 1-cell $I_{(A,\theta, \alpha)}$ is defined as follows. $I_{(A,\theta, \alpha)}=(I_{A_g}, l)$, where $l_{g,h}={\mbox{\rm id\,}}_{\theta_{g,h}}$, for any $g,h\in G$.

If $(A,\theta, \alpha)$ and $(B,\rho ,\beta)$ are objects in ${{\mathcal B}}[G]$, a 1-cell is a pair $(X,l )$, where $X=\{X_g:A_g\to B_g\}$ is a $G$-indexed family of 1-cells and $$\xymatrixrowsep{0.4in}
\xymatrixcolsep{0.4in}
\xymatrix{
F_{g}(A_{h}) \ar[d]_{\theta_{g,h}} \ar[rr]^{F_g(X_h)}&& F_g(B_h) \ar[d]^{\rho_{g,h}}\\
A_{gh} \ar[rr]_{X_{gh}}&& B_{gh} {\ar @{} [llu] |{\Downarrow l_{g,h}}},
}$$ is a $G\times G$-indexed family of isomorphism 2-cells, such that for all $f,g,h\in G$, $l_{1,g}=\operatorname{id}_{X_g}$ and equation $$\label{axiom-1-cell B[G]}
\xymatrixrowsep{0.4in}
\xymatrixcolsep{0.4in}
\xymatrix{
{\overline{f}}\ {\overline{g}}(A_{h}) \ar[d]_{{\overline{f}}(\theta_{g,h})} \ar[rr]^{{\overline{f}}\ {\overline{g}} (X_h)}&& {\overline{f}}\ {\overline{g}} (B_h) \ar[d]^{{\overline{f}}\rho_{g,h})} \ar[rr]^{\chi_{f,g}^0} && {\overline{fg}}(B_h) \ar[d]^{\rho_{fg,h}}\\
{\overline{f}}(A_{gh})  \ar[rrd]_{\theta_{f,gh}} 