When I was very young, I read a book from the 60's about classical mathematics. I was far too young to understand anything in it of course, but I remember it mentioned that category theory was the new thing that was threatening to become a new foundation for mathematics.

So I was wondering, has this happened? Is this happening? Has it become ubiquitous? And has it reached a point like ZFC has in that it is canonical and well understood? Or has it remained a sort of ad hoc merging of different areas ostensibly to allow one to carry results across from one area to another?

This makes it sound like I can understand natural transformations better by seeing what they inherit from the category of functors. Or another example, by looking at what functors inherit in the category (2-category?) of small categories, this should help me to better understand functors. Does category theory have this trickle-down structure? Or does it perhaps have a trickle-up structure? I ask because of sentences like this next quote:

There are people who want to replace ZFC with other, more category theory friendly, typed set theories (look up ETCS for example). I don't think there is an advantage, excapt making set theory more category theory friendly. For me, they don't capture properly the intuition behind what a set is. Apart from that, ZFC has still been studied much more thoroughly.

I've been thinking a little about this. I think I can answer my second question. Maps between objects can't be understood without looking the objects themselves, but at an abstract level of objects and arrows, there is nothing much to say. Similarly with functors, there is nothing much to say at the level of categories of objects and arrows. Inevitably, one looks upwards when discussing things at that level, like saying that a category is a topos, along with operads and a bunch of other things. It's a bit like examining the bolts in the ceiling of a passenger plane in lieu of looking out the window. But if we close the curtains, what else is there to talk about?

Rubi, thanks for the information, I've looked around but what what I could find is rather sketchy. Given that this is all 50+ years old, I think one must conclude that there is little interest in making a foundation out of category theory. Probably this is for the best.