0P6A Proof. We have E1λ∘ρE2∘F1σ=E_{1}\lambda\circ\rho E_{2}\circ F_{1}\sigma= =E1E2F1ε1∘E1λF1E1∘F1E12F1λE1∘F1τ1F12E2E1∘F1E1η1F1E2E1∘F1η1E2E2\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}\lambda F_{1}E_{1}\circ F_{1}E_{1}^{2}F_{1}\lambda E_{1}\circ F_{1}\tau_{1}F_{1}^{2}E_{2}E_{1}\circ F_{1}E_{1}\eta_{1}F_{1}E_{2}E_{1}\circ F_{1}\eta_{1}E_{2}E_{2} =E1E2F1ε1∘E1λF1E1∘F1E12F1λE1∘F1E12τ1E2E1∘F1E1η1F1E2E1∘F1η1E2E2\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}\lambda F_{1}E_{1}\circ F_{1}E_{1}^{2}F_{1}\lambda E_{1}\circ F_{1}E_{1}^{2}\tau_{1}E_{2}E_{1}\circ F_{1}E_{1}\eta_{1}F_{1}E_{2}E_{1}\circ F_{1}\eta_{1}E_{2}E_{2} =E1E2F1ε1∘E1λF1E1∘E1F1λE1∘E1τ1E2E1∘η1F1E2E1∘ε1F1E2E1∘F1η1E2E2\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ E_{1}\tau_{1}E_{2}E_{1}\circ\eta_{1}F_{1}E_{2}E_{1}\circ\varepsilon_{1}F_{1}E_{2}E_{1}\circ F_{1}\eta_{1}E_{2}E_{2} =E1E2F1ε1∘E1λF1E1∘E1F1λE1∘E1τ1E2E1∘η1F1E2E1\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ E_{1}\tau_{1}E_{2}E_{1}\circ\eta_{1}F_{1}E_{2}E_{1} =E1E2F1ε1∘E1E2τ1E1∘E1λF1E1∘E1F1λE1∘η1F1E2E1\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}E_{2}\tau_{1}E_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ\eta_{1}F_{1}E_{2}E_{1} =E1E2F1ε1∘E1E2F1ε1E1F1∘E1E2F12E1η1∘E1E2τ1E1∘E1λF1E1∘E1F1λE1∘η1F1E2E1\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}E_{2}F_{1}\varepsilon_{1}E_{1}F_{1}\circ E_{1}E_{2}F_{1}^{2}E_{1}\eta_{1}\circ E_{1}E_{2}\tau_{1}E_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ\eta_{1}F_{1}E_{2}E_{1} =E1E2F1ε1∘E1E2F1ε1E1F1∘E1E2τ1E12F1∘E1E2F12E1η1∘E1λF1E1∘E1F1λE1∘η1F1E2E1\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}E_{2}F_{1}\varepsilon_{1}E_{1}F_{1}\circ E_{1}E_{2}\tau_{1}E_{1}^{2}F_{1}\circ E_{1}E_{2}F_{1}^{2}E_{1}\eta_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ\eta_{1}F_{1}E_{2}E_{1} =E1E2F1ε1∘E1E2F1ε1E1F1∘E1E2F12τ1F1∘E1E2F12E1η1∘E1λF1E1∘E1F1λE1∘η1F1E2E1\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}E_{2}F_{1}\varepsilon_{1}E_{1}F_{1}\circ E_{1}E_{2}F_{1}^{2}\tau_{1}F_{1}\circ E_{1}E_{2}F_{1}^{2}E_{1}\eta_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ\eta_{1}F_{1}E_{2}E_{1} =σF1∘E2ρ∘λE1.\displaystyle=\sigma F_{1}\circ E_{2}\rho\circ\lambda E_{1}. We have ρF1∘F1ρ∘τ1E1\displaystyle\rho F_{1}\circ F_{1}\rho\circ\tau_{1}E_{1} =ε1E1F12∘F1ε1E12F12∘τ1E13F12∘F12E1τ1F12∘F12E12η1F1∘F12τ1F1∘F12E1η1\displaystyle=\varepsilon_{1}E_{1}F_{1}^{2}\circ F_{1}\varepsilon_{1}E_{1}^{2}F_{1}^{2}\circ\tau_{1}E_{1}^{3}F_{1}^{2}\circ F_{1}^{2}E_{1}\tau_{1}F_{1}^{2}\circ F_{1}^{2}E_{1}^{2}\eta_{1}F_{1}\circ F_{1}^{2}\tau_{1}F_{1}\circ F_{1}^{2}E_{1}\eta_{1} =ε1E1F12∘F1ε1E12F12∘F12τ1E1F12∘F12E1τ1F12∘F12E12η1F1∘F12τ1F1∘F12E1η1\displaystyle=\varepsilon_{1}E_{1}F_{1}^{2}\circ F_{1}\varepsilon_{1}E_{1}^{2}F_{1}^{2}\circ F_{1}^{2}\tau_{1}E_{1}F_{1}^{2}\circ F_{1}^{2}E_{1}\tau_{1}F_{1}^{2}\circ F_{1}^{2}E_{1}^{2}\eta_{1}F_{1}\circ F_{1}^{2}\tau_{1}F_{1}\circ F_{1}^{2}E_{1}\eta_{1} =ε1E1F12∘F1ε1E12F12∘F12(τ1E1∘E1τ1∘τ1E1)F12∘F12E12η1F1∘F12E1η1\displaystyle=\varepsilon_{1}E_{1}F_{1}^{2}\circ F_{1}\varepsilon_{1}E_{1}^{2}F_{1}^{2}\circ F_{1}^{2}(\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ\tau_{1}E_{1})F_{1}^{2}\circ F_{1}^{2}E_{1}^{2}\eta_{1}F_{1}\circ F_{1}^{2}E_{1}\eta_{1} =ε1E1F12∘F1ε1E12F12∘F12(E1τ1∘τ1E1∘E1τ1)F12∘F12E12η1F1∘F12E1η1\displaystyle=\varepsilon_{1}E_{1}F_{1}^{2}\circ F_{1}\varepsilon_{1}E_{1}^{2}F_{1}^{2}\circ F_{1}^{2}(E_{1}\tau_{1}\circ\tau_{1}E_{1}\circ E_{1}\tau_{1})F_{1}^{2}\circ F_{1}^{2}E_{1}^{2}\eta_{1}F_{1}\circ F_{1}^{2}E_{1}\eta_{1} =ε1E1F12∘F1ε1E12F12∘F12(E1τ1∘τ1E1)F12∘F12E13τ1∘F12E12η1F1∘F12E1η1\displaystyle=\varepsilon_{1}E_{1}F_{1}^{2}\circ F_{1}\varepsilon_{1}E_{1}^{2}F_{1}^{2}\circ F_{1}^{2}(E_{1}\tau_{1}\circ\tau_{1}E_{1})F_{1}^{2}\circ F_{1}^{2}E_{1}^{3}\tau_{1}\circ F_{1}^{2}E_{1}^{2}\eta_{1}F_{1}\circ F_{1}^{2}E_{1}\eta_{1} =E1τ1∘ρF1∘F1ρ.\displaystyle=E_{1}\tau_{1}\circ\rho F_{1}\circ F_{1}\rho. ∎