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 \$$