Marius

It took a while but I think I finally got what you are saying. The "best" one is the natural transformation \\(\alpha: LP \rightarrow RP\\) without applying it to individual objects and morphisms since if we apply it, they would be instances and therefore are themselves objects and morphisms not natural transformations. The naturality squares it needs to satisfy are the two squares you have shown : \\(RP(f_x,f_y,fz) \alpha_{XYZ} = \alpha_{XYZ}LP(f_x,f_y,f_z)\\) and \\(LP(f_x,f_y,fz) \alpha_{XYZ}^{-1} = \alpha_{XYZ}^{-1}RP(f_x,f_y,f_z)\\).

Thanks!

It took a while but I think I finally got what you are saying. The "best" one is the natural transformation \\(\alpha: LP \rightarrow RP\\) without applying it to individual objects and morphisms since if we apply it, they would be instances and therefore are themselves objects and morphisms not natural transformations. The naturality squares it needs to satisfy are the two squares you have shown : \\(RP(f_x,f_y,fz) \alpha_{XYZ} = \alpha_{XYZ}LP(f_x,f_y,f_z)\\) and \\(LP(f_x,f_y,fz) \alpha_{XYZ}^{-1} = \alpha_{XYZ}^{-1}RP(f_x,f_y,f_z)\\).

Thanks!