Functor between Order Categories

Theorem
Let $\left({S, \preceq}\right)$ and $\left({T, \preceq'}\right)$ be posets.

Let $\mathbf S$ and $\mathbf T$ be their associated poset 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:


 * $Fa \to Fb$

in $\mathbf T$ as well, i.e.:


 * $Fa \preceq' Fb$

Hence the result.