"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 $\rho_\epsilon$?