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.
Given two objects $A$ and $B$, there is an associated set/class (depending on your definition) $\operatorname{Hom}\_\mathcal{C}(A,B)$ of morphisms $A \to B$. An arrow is just such a morphism, i.e. an element of this set/class.