Nice answer for **Puzzle 280** Anindya! I had just worked out something similar based on your earlier equation, but you beat me to it. I'll post my diagrams anyway for an additional perspective. (Note that I reverted back to \\(f\\) and \\(g\\))
You showed that the top square of this cube commutes:
The sides commute because of the naturality conditions and because in our case \\(\lambda_I = \rho_I\\)
Where \\(UR(x,-) = x \\) and \\(UL(-,x) = x \\)
Since the top and sides of the square commute, so does the bottom.