Here's a category-theoretic perspective. (Check out the rest of the lectures and the associated free textbook.)

My next series of posts will be directly about the Yoneda lemma, which basically tells us that everything you could want to know about an object is contained in the morphisms going into/out of the object. Moreover, we get this knowledge in a "natural" way that makes life really easy. It's a pretty cool theorem.

In the end, we don't really care about sets at all. They're just bags with stuff in them. Who cares about bags? But we do care about functions—we want those to be rule-based. We need functions to go "from" somewhere and "to" somewhere. Let's call those things sets. Then we need these "sets" to be rule-based.

I'm grateful for your comments. They're very useful, and you raise good points. I've got most of a post already about how functions give meaning to the elements of sets. As for how functions is a verb, think of properties as existing in verbs. So to know something, you need to observe it in some way, which means it has to affect your sensory devices, such as your ears, eyes, thermometers, whatever. You know dogs, for example, by the way they bark, by the way they lick, they way they look, etc. So properties exist in the verbs. "Legs" are a noun, but all of your knowledge about them has to come from verbs. Does that make sense?

You raise a good point. Think of category theory as a language for expressing, in this case, the logic of sets and functions. You still need to know what that logic is. Then you can use category theory to work efficiently with that logic owing to its general-abstract nature.

I agree. I'm shifting gears to work on something basically aimed at the idea that the intelligent layperson can grasp Yoneda lemma and adjunction if it's explained.

It's older terminology. Everyone says image now.

I thought I had it right, and then mixed it up in my head myself.

Steam is run out of. This was poorly conceived to begin with, arrogant in its inherent design, and even I don't have the patience for it anyway. I'll do a series about adjunction directly and Yoneda as well.

Honestly my real justification would be "adjoint functors awesome, and you need categories to do adjoint functors, so use categories." More broadly...as long as it's free to create a category out of whatever you're studying, there's clearly no harm. The question is whether anything's lost by treating the subject as a category, and while I fully expect that there are entire universes of mathematics and reality out there where categories are harmful, I don't think we live in one like that. Categories may not capture everything you can think of, but they can capture so much that I'd be stunned if they didn't yield amazing fruit eventually. I'd acknowledge that novel, groundbreaking theorems are still forthcoming.