\ar[rr]_{{\overline{f}}(X_{gh})}&& {\overline{f}}(B_{gh}) {\ar @{} [d] |{\Downarrow l_{f,gh}}} {\ar @{} [llu] |{\Downarrow {\overline{f}}(l_{g,h})}} \ar[rr]_{\rho_{f,gh}} && {\ar @{} [llu] |{\Downarrow \beta_{f,g,h}}} B_{fgh}\\
&& A_{fgh} \ar[rru]_{X_{fgh}} &&
}$$

$$\xymatrixrowsep{0.4in}
\xymatrixcolsep{0.4in}
\xymatrix{
{\overline{f}}\ {\overline{g}}(A_{h}) \ar[d]_{{\overline{f}}(\theta_{g,h})} \ar[rrd]_{\chi^0_{f,g}} \ar[rr]^{{\overline{f}}\ {\overline{g}} (X_h)}&& {\overline{f}}\ {\overline{g}} (B_h) {\ar @{} [d] |{\Downarrow (\chi_{f,g})_{X_h}}} \ar[rr]^{\chi_{f,g}^0} && {\overline{fg}}(B_h) \ar[d]^{\rho_{fg,h}}\\
{\overline{f}}(A_{gh})  \ar[rrd]_{\theta_{f,gh}} {\ar @{} [rr] |{\Downarrow \alpha_{f,g,h}}} && {\overline{fg}}(A_{h}) \ar[rru]_{{\overline{fg}}(X_h)} \ar[d]_{\theta_{fg,h}} {\ar @{} [rr] |{\Downarrow l_{fg,h}}} && \\
&& A_{fgh} \ar[rru]_{X_{fgh}} &&
}$$ holds in ${{\mathcal B}}(F_f(F_g (A_h)),B_{fgh})$. If $(X,l),$ $(Y,s)$ are 1-cells, a 2-cell $m:(X,l)\Rightarrow (Y,s)$ is a $G$-indexed family of 2-cells $m=\{m_g:X_g\to Y_g\}$ such that for all $g,f \in G$, equation

$$\label{axiom-2-cell B[G]}
\xymatrixrowsep{0.4in}
\xymatrixcolsep{0.5in}
\xymatrix{
F_{g}(A_{h}) \ar[d]_{\theta_{g,h}} \ar@/^2pc/[rr]^{F_g(X_h)}&& F_g(B_h) \ar[d]^{\rho_{g,h}} {\ar @{} [ll] |{\Downarrow l_{g,h}}} \\
A_{gh} \ar@/^2pc/[rr]_{X_{gh}} \ar@/_2pc/[rr]_{Y_{gh}} && B_{gh} {\ar @{} [ll] |{\Downarrow m_{gh}}}
}$$

$$\xymatrixrowsep{0.4in}
\xymatrixcolsep{0.5in}
\xymatrix{
F_{g}(A_{h})  {\ar @{} [rr] |{\Downarrow F_g(m_h)}}\ar[d]_{\theta_{g,h}} \ar@/^2pc/[rr]^{F_g(X_h)} \ar@/_2pc/[rr]^{F_g(Y_h)} && F_g(B_h) \ar[d]^{\rho_{g,h}}\\
A_{gh}  \ar@/_2pc/[rr]_{Y_{gh}} && {\ar @{} [ll] |{\Downarrow s_{g,h}}} B_{gh} 
}$$ holds in ${{\mathcal B}}(F_g(A_h),B_{gh})$.

The (vertical) composition in each category ${{\mathcal B}}[G]((A,\theta, \alpha),(B,\rho ,\beta))$ is defined pointwise.

Now, let us define the horizontal composition $\circ:{{\mathcal B}}[G]((A,\theta, \alpha),(B,\rho ,\beta))\times {{\mathcal B}}[G]((C,\kappa,\gamma),(A,\theta, \alpha))\to {{\mathcal B}}[G]((C,\kappa,\gamma),(B,\rho ,\beta)).$ If $(A,\theta, \alpha)$ and $(B,\rho ,\beta)$ are 0-cells, and $$(X,l) \in {{\mathcal B}}[G]((A,\theta, \alpha),(B,\rho ,\beta)),\,\, 