which is presented by generators and equations. The monoid ${\cal L}_\omega$ is just another presentation by generators and equations of ${\cal
S}^\ast$. A reason for introducing it in [@DP03] was to make simpler reduction to normal form, without being encumbered by the sources and targets of arrow terms. We presuppose the reader is acquainted with ${\cal L}_\omega$, but indications about how this monoid is presented will be given below when we deal with composition in *Frobse*. This monoid interests us here only as an auxiliary, leading towards the geometric categories *Frz* and *Frobse*, which we will consider below.

Out of the material of the monoid ${\cal S}^{\ast}$ we return to $\cal S$, by building a category isomorphic to $\cal S$, in the following way. Let ${\cal S}^{\ast t}$ be the category whose objects are the natural numbers, and whose arrows are the triples ${\langle [f], n, m\rangle}$ such that there is an arrow ${g:\!n\str m}$ in $[f]$; the arrow ${\langle [f], n, m\rangle}$ is of *type* $n\str m$ in ${\cal S}^{\ast t}$, which means that its source is $n$ and its target $m$. The composition of ${\langle [f_1], n, m\rangle}$ and ${\langle [f_2],m,k\rangle}$ is defined as ${\langle [f_2][f_1],n,k\rangle}$, and the 