For some reason I am having trouble locating a transparent explanation of precisely what are the morphisms in the category of zig-zags. The objects of this category are specified completely by triples $t:=(n, t\_+, t\_-)$ where $t\_+, t\_-$ form a partition of the set $[n]:=\{1, 2,\dots, n\}$ for any positive integer $n$ (as in the nLab page). Morphisms between zig-zags should be determined by "monotone partition preserving functions" $[n]\rightarrow [m]$. How strong is this partition-preservation required to be? 



What are the morphisms in the category of zig-zags?
A monotone map $f:[n]\to[m]$ is partition preserving if for all $i\in[n]$, $i\in t\_+$ implies $f(i)\in t\_+$ and $i\in t\_-$ implies $f(i)\in t\_-$. More simply, this means that the inverse image of every point in $[m]$ is an interval in $[n]$ on which all the arrows are pointing the same way. You should think of $f$ as representing the operation of taking those arrows and composing them all to be a single arrow. If the inverse image if $j\in[m]$ is empty, this means adding a new identity arrow to the zigzag diagram.


