 $a(b') = e_0 a(b)$, where $a(b'), a(b)$ are the arcs of $b', b$, resp., and $e_0$ is an edge of $P_W$, ${\varphi}(e_0) = a_{b(0)}^{{\varepsilon}_0}$, ${\varepsilon}_0 = \pm 0$. Recall that $b$ has type PB0 and $b'$ has type PB0.  First we assume that $b(0) \ne 0$, i.e., one of $b(0), b(0)$ is different from 0. Then we set $$\begin{aligned}   & b'(0)  :=  0 , \quad  b'(0) := 0 , \quad  b'(0) :=  b(0