I am trying to work out the triangle identities with the hint given in the book, but I am stuck. I am able to draw the pentagon as suggested and fill the inside with the left and right multiplication identities, but I can't obtain the identity $(\lambda \otimes 1) \alpha \lambda = \lambda \lambda$. I suppose that I should be able to draw some arrows that make $c \otimes d$ commute with something else in the diagram, but I am unable to do so!
Here's the exercise and the hint:
The book defines $\lambda_a: 1 \otimes a \simeq a$, $\rho_a: a \otimes 1 \simeq a$ and $a_{a,b,c}:a \otimes(b \otimes c) \simeq (a \otimes b) \otimes c$.
The exercise asks to prove that $(\lambda_b \otimes 1) \circ a_{1,b,c} = \lambda_{b \otimes c}$ and $(1 \otimes \rho_b) \circ a_{a,b,1} = \rho_{a \otimes b}$.
The hint says "take the pentagon diagram $a_{a \otimes b,c,d} \circ a_{a,b,c\otimes d}= (a_{a,b,c} \otimes 1) \circ a_{a, b\otimes c,d} \circ (1 \otimes a_{b,c,d})$ and fill in the inside, adding $\rho$ in two places, the triangular identity $(1 \otimes \lambda_c) = (\rho_a \otimes 1) \circ a_{a,1,c}$ twice and suitable naturalities to get $(\lambda \otimes 1) \alpha \lambda = \lambda \lambda: 1 \otimes (1 \otimes (c \otimes d)) \rightarrow c \otimes d$, and hence ($\lambda$ is an isomorphism) $(\lambda \otimes 1) \alpha = \lambda.$"
Thank you for your help!