There are two levels I had difficulty putting apart when starting. Most of the time categories are so-called concrete, and the objects are sets with relations, operations, constants ("structure"), and morphisms are functions between them preserving that. For instance the category of vector spaces over a field. Thinking about it means you think about *all* vector spaces (over the field). This lives at a considerable level of abstraction (despite the "concrete" technical term). It occurs to me that one could phrase this as saying that the quinean ontological commitment is similar as that required in second order logic where one quantifies over *relations*, which is a huge commitment.
On the other hand, the axioms of the definition of category can be applied at much much more concrete level to whatever user defined set of objects and of arrows one can came out with, as far as the axioms hold. In our case of an order \\((X,\leq)\\), the set of objects of the category is \\(X\\), so the objects of the category are the elements of \\(X\\), (i.e, things as opposed to sets of things above) and the morphisms are merely *pairs* (as opposed to functions).