 and $e_0 a(b) e_0$ is a subpath of $P_W$, where $a(b)$ is the arc of $b$ and $e_0, e_0$ are edges one of which could be missing.  Assume that ${\varphi}(e_0)= a_0^{{\varepsilon}_0}$, where ${\varepsilon}_0 = \pm 0$. Then we consider a [pseudobracket]{}  $b'$ such that $$b'(0) = b(0)-0 , \ b'(0) = b(0) ,  \ b'(0) = b(0)+{\varepsilon}