follows from the symmetry of the Kronecker delta, or, alternatively, by the reality of $\delta_{mm'}$ and the rules for complex conjugation.

The $j$ labels on the edges of the spin networks indicate which carrier space $\CS_j$ the ket or bra lies in, or in which carrier space a contraction has taken place. If there are distinct carrier spaces with the same $j$ label, then additional distinguishing information must be supplied.

Intertwiners
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An $SU(2)$ intertwiner (the only kind we are interested in) is a linear map between two vector spaces that commutes with the action of $SU(2)$ on the two spaces. The only case of interest here is where the target space is $\Complexes$, consisting of scalars, that is, invariants under rotations. The Wigner intertwiner $W:\CS_{j_1}\otimes\CS_{j_2 }\otimes\CS_{j_3} \to \Complexes$ is of this type. But a linear map from a Hilbert space to $\Complexes$ can be thought of as a dual or bra vector, for example, we can associate the map $W$ with a bra vector $\bra{W}$ belonging to $\CS^*_{j_1}\otimes\CS^*_{j_2}\otimes\CS^*_{ j_3}$. The spin network notation for $\bra{W}$ is shown in Fig. \[Wintertwiner\]. The components of the Wigner intertwiner, that is, the $3j$-symbol, are obtained by inserting basis kets, the ket chevron 