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: 
I recall a really fun talk at a topology conference sometime in the late 1980s/ early 1990s. The speaker said "we're going to take as objects in a category the set of finite collections of evenly-distributed points on the $x$-axis between $x = 1$ and $x = 2$ -- we might have 3 points or 12 points or ...whatever."

OK, so far I was following him. Then he said "and a *morphism* in this category will be a braid arranged thus":

[![enter image description here](https://i.stack.imgur.com/Uyici.png)](https://i.stack.imgur.com/Uyici.png)

where the locations and directions of over- and under-crossings was arbitrary. He then went on to talk about natural transformations from this category to various others, developing knot polynomials in a remarkably abstract way. I have no idea whether anyone still does this sort of thing, but at the time, it was kinda mindblowing to see something so very concrete as a 'morphism'. I leave it to you to guess how composition of morphisms was defined...
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.