What exactly is an arrow in a category?

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!

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