speaking, a frieze is a tangle without crossings in whose regions we find circular forms that correspond bijectively to the ordinals contained in the infinite ordinal $\varepsilon_0$. In [@DP03] one can found a proof that ${\cal L}_\omega$ is isomorphic to the monoid of friezes, and from that the isomorphism of the categories ${\cal L}^{\cal A}_\omega$ and *Frz* follows. So, by Proposition 2, the categories *Frob* and *Frz* are isomorphic. By this last isomorphism, the arrows on the left are mapped to the friezes on the right, with the type associated to the friezes being those of the arrows:

(200,60)(-60,0)

(-120,25)[(0,0)\[l\][${\varepsilon^\Box_n\!:n\pl 1\str
n}$]{}]{}

[ (0,10)[(1,0)[200]{}]{} (0,10)[(0,1)[30]{}]{} (0,40)[(1,0)[200]{}]{}]{}

(30,0)[(0,0)\[b\][$1$]{}]{} (90,0)[(0,0)\[b\][$2n\pl 1$]{}]{} (30,45)[(0,0)\[b\][$1$]{}]{} (90,45)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,45)[(0,0)\[b\][$2n\pl 2$]{}]{} (150,45)[(0,0)\[b\][$2n\pl 3$]{}]{}

(30,10)[(0,1)[30]{}]{} (90,10)[(2,1)[60]{}]{}

(105,40)[(30,30)\[b\]]{}

(50,1)[$\ldots$]{} (50,46)[$\ldots$]{} (147,25)[$\ldots$]{}

(200,70)(-60,0)

(-120,25)[(0,0)\[l\][${\delta^\Box_n\!:n\pl 1\str n\pl
2}$]{}]{}

[ (0,10)[(1,0)[200]{}]{} (0,10)[(0,1)[30]{}]{} (0,40)[(1,0)[200]{}]{}]{}

(30,0)[(0,0)\[b\][$1$]{}]{} (90,0)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,0)[(0,0)\[b\][$2n\pl 2$]{}]{} (150,0)[(0,0)\[b\][$2n\pl 3$]{}]{} (180,0)[(0,0)\[b\][$2n\pl 4$]{}]{}

(30,45)[(0,0)\[b\][$1$]{}]{} (90,45)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,45)[(0,0)\[b\][$2n\pl 2$]{}]{}

(30,10)[(0,1)[30]{}]{} (90,10)[(0,1)[30]{}]{} (180,10)[(-2,1)[60]{}]{}

(135,10)[(30,30)\[t\]]{}

(55,25)[$\ldots$]{} (177,25)[$\ldots$]{}

(200,70)(-60,0)

(-120,25)[(0,0)\[l\][${\varepsilon^\Diamond_n\!:n\str
n\pl 1}$]{}]{}

[ (0,10)[(1,0)[200]{}]{} (0,10)[(0,1)[30]{}]{} (0,40)[(1,0)[200]{}]{}]{}

(30,45)[(0,0)\[b\][$1$]{}]{} (90,45)[(0,0)\[b\][$2n\pl 1$]{}]{} (30,0)[(0,0)\[b\][$1$]{}]{} (90,0)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,0)[(0,0)\[b\][$2n\pl 2$]{}]{} (150,0)[(0,0)\[b\][$2n\pl 3$]{}]{}

(30,10)[(0,1)[30]{}]{} (90,40)[(2,-1)[60]{}]{}

(105,10)[(30,30)\[t\]]{}

(50,1)[$\ldots$]{} (50,46)[$\ldots$]{} (147,25)[$\ldots$]{}

(200,70)(-60,0)

(-120,25)[(0,0)\[l\][${\delta^\Diamond_n\!:n\pl 2\str
n\pl 1}$]{}]{}

[ (0,10)[(1,0)[200]{}]{} (0,10)[(0,1)[30]{}]{} (0,40)[(1,0)[200]{}]{}]{}

(30,45)[(0,0)\[b\][$1$]{}]{} (90,45)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,45)[(0,0)\[b\][$2n\pl 2$]{}]{} (150,45)[(0,0)\[b\][$2n\pl 3$]{}]{} (180,45)[(0,0)\[b\][$2n\pl 4$]{}]{}

(30,0)[(0,0)\[b\][$1$]{}]{} (90,0)[(0,0)\[b\][$2n\pl 1$]{}]{} (120,0)[(0,0)\[b\][$2n\pl 2$]{}]{}

(30,10)[(0,1)[30]{}]{} (90,10)[(0,1)[30]{}]{} 