I am trying to learn category from some lecture notes from my university. I just came across the statement:
If $P$ and $Q$ are partially ordered sets, regarded as categories, a functor $P \to Q$ is just an order-preserving map.
I don't understand how to see that this statement is true! I am, in general, having some trouble grasping the concept of one object categories, too. Any help on how to best understand those concepts would be much appreciated!
Don't think of posets as one-object categories. Posets can be regarded as categories where every element of the poset is an object in the category, and an arrow $A \to B$ in the category is precisely an instance of the fact that $A \leq B$. (So there is only ever at most one arrow from $A$ to $B$.)
Then a functor $F: \mathcal{P} \to \mathcal{Q}$ satisfies the property that whenever $A, B \in \mathcal{P}$ are such that there is a [necessarily unique] arrow $A \to B$, there is a [necessarily unique] arrow $FA \to FB$. Translated into poset-speak, that means that whenever $A \leq B$, it is true that $FA \leq FB$. That is, if $F$ is a functor then it is an order-preserving map.
The converse is also true for the same reasons.