Functor between Order Categories

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Theorem

Let $\struct {S, \preceq}$ and $\struct {T, \preceq'}$ be ordered sets.

Let $\mathbf S$ and $\mathbf T$ be their associated order categories, respectively.

Let $F: \mathbf S \to \mathbf T$ be a functor.


Then its object functor $F: S \to T$ is a monotone mapping.


Proof

Suppose that for some $a, b \in S$, we have:

$a \preceq b$

Then there is a morphism $a \to b$ in $\mathbf S$.


As $F$ is a functor, it follows that there is a morphism:

$F a \to F b$

in $\mathbf T$ as well, that is:

$F a \preceq' F b$


Hence the result.

$\blacksquare$


Sources