Michael wrote in 37:
> The last two questions are still unclear to me...

Remember that in the case of a natural transformation \\(\alpha: F \rightarrow G \\) you have two functors \\(F, G : \mathcal{C} \rightarrow \mathcal {D}\\) and that for objects \\(A\\) and \\(B\\) in \\(\mathcal{C}\\) and morphism \\(f : A \rightarrow B\\) we need to have \\(\alpha_B \circ F(f) = G(f) \circ \alpha_A\\)

In this case we have functors \\(LP, RP: \mathcal{C} \times \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C} \\) and we need to show that the following sqaures commute:


or with the functors applied:


Also, remember how the product functor acts on morphisms:


With these hints and your projection diagram, you should be able to solve the problem.