Let $\mathbf{T}$ be the category defined as follows:

An object is given by a triple of data $t \coloneqq (n,t_+,t_-)$, which is a partition of the poset$\{i \;|\; 1\leq i\leq n\}$ for some natural number$n \geq 0$ into two subset parts $t_+$ and $t_-$. (The corresponding diagram as above has nodes $t_0, \ldots, t_n$, and the arrow between $t_{i-1}$ and $t_i$ points forward to $t_i$ if $i \in t_+$, and backward if $i \in t_-$.)

We will call $\mathbf{T}$ the zigzag category, or the category of zigzag types.

Also, notice that we have cleverly hidden the empty set among the objects. We pat ourselves on the backs for doing this. (Here the zigzag type consists of a single node.)