but one that makes our overall notation for mapping kets to bras and vice versa consistent (see Fig. \[Kconsist\] below). As indicated, we abbreviate the result by taking the original network for the bra $\bra{\phi}$ and simply flipping the direction of the chevron. We do not unstar the identifying symbol. As indicated, the result differs from Hermitian conjugation applied to $\bra{\phi}$, which is the ket $\ket{\phi}$ (without the star).

The requirement that $K_1^{-1}$ actually be the inverse of $K_1$ is illustrated in Fig. \[K1invreq\]. The stub on the network for $K^{-1}$ is inverted so that the output of the first step is fed into the first operand of $K^{-1}$.

The identity represented by Fig. \[K1invreq\] is usually encountered in practice in the form shown in Fig. \[2stubs\] (a fragment of a spin network). The $2j$-node inverts the direction of the arrow. Two such inversions, with stubs pointing in opposite directions, annihilate one another.

We have made an independent definition of the spin network in Fig. \[Kinv\], but it is the same as the spin network for $K$, shown in Fig. \[2jKdef\], with both bra chevrons flipped. Since we now have a convention for flipping bra chevrons (by applying $K_1^{-1}$), for 