What should my internal representation of an object in the category of small categories be?

Question

What exactly should I think of a category as? The usual definition of a category as a collection of objects + some morphisms between these objects seems a little vague to me.

Say that we have $\textbf{Set}$, the category of sets and functions and a category that I made up $\textbf{Set}_{id}$ which is the category where objects are sets and the morphisms are only the identity morphisms. Is the one I made up a real category and if so, how is it related to $\textbf{Set}$?

Now let's consider the category $\textbf{2}$ where there are two objects $A$ and $B$, their identity morphisms, and a single morphism $f : A \to B$. What does this $f$ morphism do to the elements of $A$?

Answer 1

Morphisms in categories don't do anything to elements of anything, in general. In your category $A$ might be Richard Nixon, $B$ the complex numbers, and $f$ my coffee mug. You introduce identity arrows, define a composition in the only possible way and you have a category. $f$ need not be a function or anything like a function.

The problem here may be less with the definition of a category and more with your not having worked through enough examples. You also may not have seen a precise definition of a category, so make sure to look one up. Do you know how every group becomes a category? Every partially ordered set? You should also check yourself whether your proposed category of sets and identity morphisms is actually a category. There is a short list of axioms that are easy to verify. If you know how to prove a set with a given operation is a group, you should have no trouble with this.

Answer 2

Yes, $\textbf{Set}_{id}$ is, in fact, a category, and it is the smallest subcategory of $\textbf{Set}$ that share its objects.

To your second point, why does it have to do anything? Look at the elements of sets in usual mathematics: do they necessarily have an internal structure? No, because if we want to add abstraction to our study, it is necessary not to assume structure from this elements.

The same applies to a morphism: it does not have to do anything; and $A$ does not need to have internal structure either. Just think of $A$ and $B$ as points and $F$ as literally an arrow connecting $A$ to $B$, in this order: it is your category, or at least one interpretation of it, and $f$ does not affect the elements of $A$, which in turn does not have elements!