[Twisol]

> Presumably objects (since beer mugs are objects in the conventional sense), but then what are the arrows?

This is definitely one of the most unintuitive parts of early category theory. The "objects" are not automatically the things you're studying; often they're best modeled as the arrows. It took me a really long time to internalize that.

Put it this way: the arrows are the actually interesting things, from a categorical perspective. The objects literally only exist to tell you which arrows you can fit together. The objects don't contain any other information. We often say that sets are objects, or topological spaces are objects, but that's only because we're studying functions or homeomorphisms, and if we want to chain those together, we need to make sure that one ends where the other begins or else the composition isn't well-defined.

No, we don't expect everything to be modeled directly as a category. That's not how the abstraction pays for itself.