Question: The usual definition of a category states: a category $\mathbf{C}$ consists of:

1. A collection $\text{ob}(\mathbf{C})$ of *objects*
2. A collection $\text{arr}(\mathbf{C})$ of *arrows*
3. Some rules on the behaviour of these two types of objects

Leaving aside what *collection* here means, I realized that I have never seen a clear definition of what an arrow in a category is.
Surely, in the typical categories like $\textbf{Set}$, $\textbf{Top}$ or $\textbf{Grp}$ arrows are functions that ... But there are categories whose arrows are not "functions".

If you have a poset $(P,\le)$ then you can get a category whose objects are elements of $P$ and such that there is an arrow $x\to y$ if and only if $x\le y$ in $P$. Okay but what kind of "entity" is the arrow there?, is there a precise, rigorous definition or do just have to accept that there are objects and arrows and asume that they follow the required rules, just like we asume there some things called "numbers" that follow the rules of arithmetic?

I hope I made myself clear

Thanks!

Answer: 
An arrow is an element of the set of arrows. The set of arrows is any set you want.

Let me explain by talking about directed graphs. You can imagine a category as a directed graph with an associative composition law. A directed graph is something with a set of vertices $V$ and a set of directed arcs $A$ together with a source and target function $s, t : A \to V$. Now normally we do something like set $V = \{1,\dots,n\}$ and $A \subseteq V \times V$ and $s(a,b) = a$ and $t(a,b) = b$. This is a common encoding, but there's nothing in the definition of a directed graph that says that $A$ must be a set of ordered pairs and $s$ and $t$ must be the first and second coordinates of those pairs. I could just as well have $V = \{1,2,3,4\}$ and $A = \{1,2,3\}$ with $s(i) = i$ and $t(i) = i + 1$. This also defines a directed graph.

So for categories, the set of arrows is literally just any set and any element of that set is called an arrow. Maybe for convenience you want your arrows encoded as ordered pairs but the definition doesn't force any particular encoding on you.

P.S. Actually you probably don't want your arrows to be ordered pairs because often ${\rm Hom}(A,B)$ has multiple elements whereas there is only one ordered pair $(A,B)$. Maybe for posets this is an OK encoding though.
To respond to your first remark, the collections $ob(C)$ and $arr(C)$ are what we call $\textbf{classes}$. Naively, this means that we can think of them each as a "set," except that they do not necessarily obey the axioms that are necessary to describe them as a set in the formal sense. If this feels wishy-washy to you, that is because it is. For a more satisfactory definition, you might be interested in looking through *Categories for the Working Mathematician*, by Mac Lane. 

From this perspective, a particular arrow $f\in \text{Hom}(A,B)$ is just an element of the class of the arrows from $A$ to $B$. The category in question should define precisely what exactly this class is comprised of, but in general, an arrow is not a particularly sophisticated "entity;" it is just an element of this class.

For categories which we call $\textbf{small}$, the classes $ob(C)$ and $arr(C)$ are actually sets, and then an arrow is just an element of the set $arr(C)$ which satisfies the formal properties that are outlined in the definition of a category.