Maybe a more direct answer is that in the very next post in the series, we’ll see that sets can be considered the objects of the category of sets and functions, and also the objects of the category of sets and binary relations. Functions are binary relations, so that’s not a perfect answer, but yes, you can think of an individual category as a context of sorts through which you view the objects, like how you can view a tomato as a fruit or vegetable depending on the context.
One of the reasons for my own interest in category theory is my interest in the question you raise. I’m hoping that we’ll explore the idea that universal properties offer an “objective” way of defining “subjective” categories.
Maybe a more direct answer is that in the very next post in the series, we’ll see that sets can be considered the objects of the category of sets and functions, and also the objects of the category of sets and binary relations. Functions are binary relations, so that’s not a perfect answer, but yes, you can think of an individual category as a context of sorts through which you view the objects, like how you can view a tomato as a fruit or vegetable depending on the context.