What exactly is an arrow in a category? The usual definition of a category states: a category $\mathbf{C}$ consists of:


*

*A collection $\text{ob}(\mathbf{C})$ of objects

*A collection $\text{arr}(\mathbf{C})$ of arrows

*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!

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. 
