"Pedantic remark: Aren't you missing the identit..."

Then we do it all again for all the inverses $~$x^{-1}$~$, creating the functions $~$\\rho\_{x^{-1}}$~$\.

Pedantic remark: Aren't you missing the identity of free groups in your intuitive construction?

We have the $~$\rho_x$~$ and the $~$\rho_{x^{-1}}$~$. Where is $~$\rho_\epsilon$~$?