It is well known that the augmented simplex category is the free strict monoidal category with a monoid: the monoidal product is concatenation, and the monoid is the terminal object (called [0] or Δ^{0} by geometers). Perhaps less well known is the free strict monoidal category with *k* monoids (where *k* is any cardinal, possibly infinite):

- The objects are pairs (
*n*, *P*) where *n* is a natural number (possibly zero) and *P* is a partition of the set { 0, …, *n* – 1 } into *k* disjoint subsets *P*_{j}, 0 ≤ *j* < *k*.
- The morphisms (
*n*, *P*) → (*m*, *Q*) are monotone maps { 0, …, *n* – 1 } → { 0, …, *m* – 1 } that respect the partitions, i.e. send members of *P*_{j} to members of *Q*_{j}.

In other words, it is the category of finite linear orders equipped *k*-colouring of its elements. From this point of view, the monoidal product is best interpreted as concatenation of (coloured) linear orders. The *k* monoids are the *k* different objects of the form (1, *P*). Amusingly, the monoidal product on the set of objects makes it the free monoid on *k* generators.

The opposite category, i.e. the free strict monoidal category with *k* comonoids, can be described concretely as follows:

- The objects are pairs (
*n*, *P*) as before.
- The morphisms (
*m*, *Q*) → (*n*, *P*) are monotone maps *f* : { 0, …, *m* } → { 0, …, *n* } that satisfy the following axioms:
*f*(0) = 0 and *f*(*m*) = *n*.
- If
*i* is a member of *P*_{j}, then for *f*(*i*) ≤ *e* < *f*(*i* + 1), *e* is a member of *Q*_{j} .

The above should be interpreted as the category of finite intervals equipped with a *k*-colouring of its *edges*. From this point of view, the monoidal product is the wedge sum. The special case where *k* = 2 is especially useful in relation to the problem of inverting morphisms in a category: for this, see [Dwyer and Kan, *Calculating simplicial localizations*].