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: Thanks.