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: 
$\text{Hom}\_C(A,B)$ can be literally any set you want and has nothing at all to do with functions $A \rightarrow B$, as long as you define what "composing" arrows means and there is an identity arrow for each object you have a category, your set of arrows can be literally any set you want. Same goes for your collection of objects. Just to get my point across (risking sounding ridiculous) I will give an extreme example.

I can define a category $\mathcal C$ where there are two objects, "Apple" and "Banana" and we define 

$$\text{Hom}(\text{Apple}, \text{Banana}) = \{x \in \mathbb N: \text{The word "father" is mentioned on page } x \text{ in the bible}\}$$

$$\text{Hom}(\text{Banana},\text{Apple}) = \emptyset$$

$$\text{Hom}(\text{Banana},\text{Banana}) = \text{Hom}(\text{Apple},\text{Apple}) = \{\*\}$$

We define composition in the only way possible letting $\*$ be an identity arrow for both "Apple" and "Banana".

Here an arrow in $\mathcal C$, $\text{Apple} \rightarrow \text{Banana}$ is a natural number. 

An example of a collection of objects and arrows that does not form a category (again, at the risk of sounding ridiculous) is if the objects of $\mathcal D$ is the people on earth and an arrow $X \rightarrow Y$ is a date that $X$ has shaked $Y's$ hand on. Yesterday I met my professor so there is an arrow $$\text{(27th of march 2020)}: \text{Noel} \rightarrow \text{Noel's professor}$$

Let's say that there is an arrow $X \rightarrow Y$ and an arrow $Y \rightarrow Z$ in $\mathcal D$, can we even guarantee that there exists an arrow $X \rightarrow Z$? We can't since $X$ hasn't necessarily shaked $Z's$ hand. 

Does this get my point across? I will give a more mathematical example of a category where morphisms aren't in any way functions. The category $\textbf{Set}^{op}$ has objects all the sets and we define an arrow in $\textbf{Set}^{op}$ $A \rightarrow B$ to be a function $B \rightarrow A$ in $\textbf{Set}$. That's enough examples! Hope you understood.
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...