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preorder as a category
Every preorder $P$ has an associated structure of a category. Before describing what this category is, we first associate $P$ with a simpler structure, that of a precategory.
Let’s call this $\operatorname{PreCat}(P)$. The objects of this precategory are elements of $P$ and for every $a,b\in P$, $\hom(a,b)$ is either a singleton if $a\leq b$, or the empty set otherwise. The category associated with $P$ is the category generated by enlarging $\operatorname{PreCat}(P)$. For now, call this category $\operatorname{Cat}(P)$. Then we see that the objects of $\operatorname{Cat}(P)$ are again elements of $P$, and for every $a,b\in P$, $\hom(a,b)$ is the set of all finite chains $f$ from $a$ to $b$.
With this association, we see the following constructs also have the structure of a category:

a partition of a (nonempty) set (a set with an equivalence relation): $\hom(a,b)$ is nonempty iff $a$ and $b$ belong to the same partition
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