of a sum on $m$ by a joining of edges. The general usage is shown in Fig. \[msum\]. The directions of the arrows and the stars on one of the $m$’s must be coordinated as shown for the identity to be used as shown. The small arrow in the final diagram in Fig. \[msum\] (a fragment of a spin network) is a reminder of the directions of the arrows before the sum. This small arrow is omitted in the final diagram of Fig. \[roi\] (the identity diagram) because the chevrons already indicate the direction of the edge. Recall that edges are also joined on contracting a bra with a ket, as in Fig. \[scalarprod\].

The use of the identity diagram is illustrated in Fig. \[idj\]. The first row illustrates its action on kets (as a map $:\CS_j\to\CS_j$), where bra and ket chevons are combined as in Fig. \[scalarprod\]. The small arrow is omitted after the joining of the bra and ket chevrons because the remaining chevron indicates the direction of the edge. The second row of the figure illustrates its action on bras (the map $:\CS^*_j\to\CS_j^*$), and the third row illustrates its action as a map $:\CS_j^*\otimes\CS_j\to\Complexes$, that is, the 