Poset category

The poset category of a poset $$(P, \leq)$$ is the category $$\mathsf{P}$$ with:

$$\operatorname{Ob}(\mathsf{P}) := P$$

and with $$\operatorname{Hom}_{\mathsf{P}}(X, Y)$$ containing a single element if $$X \leq Y$$ and empty otherwise.

If $$X \leq Y \leq Z$$ then the composition of the unique map $$X \to Y$$ and the unique map $$Y \to Z$$ is defined to be the unique map $$X \to Z$$.