every $l\geq 0$, while the generator $b^\alpha_{2n+1}$ will have $n\pl l\str n\pl
l\pl 1$, and the generator $c^\alpha_{2n+1}$ will have $n\pl l\str
n\pl l$. (This typing is explained by the typing of the friezes below.) The generator $a^\alpha_{2n+2}$ will have as associated types $n\pl l\pl 2\str n\pl l\pl 1$, the generator $b^\alpha_{2n+2}$ will have $n\pl l\pl 1\str n\pl l\pl 2$, and the generator $c^\alpha_{2n+2}$ will have $n\pl l\pl 1\str n\pl l\pl
1$. Multiplication of terms now becomes composition, and takes the types into account. Two typed terms of ${\cal L}_\omega$ stand for the same arrow of the category ${\cal L}^{\cal A}_\omega$ if and only if they are of the same type and equal in ${\cal L}_\omega$. We can then assert the following.

This follows immediately from the isomorphism of *Frob* and $\cal S_A$, proved in the preceding section, the isomorphism of $\cal S_A$ and ${\cal S}_{\cal A}^{\ast t}$, which follows from Proposition 1, and the isomorphism of ${\cal S}_{\cal
A}^{\ast t}$ and ${\cal L}^{\cal A}_\omega$.

From [@DP03] (Section on ${\cal L}_\omega$, ${\cal K}_\omega$ *and friezes*) one can infer that the category ${\cal
L}^{\cal A}_\omega$ is isomorphic to a category *Frz* whose arrows are diagrams called *friezes* with associated types. Roughly 