This is a pretty well-known example, but I like too much to go without mentioning it. The Gelfand representation gives an equivalence between the category of commutative, unital C*-algebras and the opposite category of compact Hausdorff spaces.

I was not worrying about the foundations of category theory for no purpose; at a certain point working with categories, it was necessary for me to fix a universe and work with that. I had never had to do this before, and it led me to look a bit more into foundations of category theory and mathematics founded only on category theory. I'm perfectly happy to stop with fixing a universe, but if there were additional advantages to be had by being more careful with category theory, then I didn't want to miss out.

I am also interested in this connection. Searching the internet and literature hasn't been too forthcoming yet, but I have only just begun. It seems to me that a likely connection would be through representations of amenable groups. Jaoby, have you found anything useful in this direction on your own?

Thanks! It seems I was confused and/or misinformed. I did not know their paper used K-Theory.

I agree with you that your answer answers the question of `why derived functors' pretty well. Thanks again for your answer, although I was hoping for something specific about the satellite.

Thanks for the answer. I'll have to think about this a bit more; I've never worked with Kan extensions before.

Yes, thanks. The different setting in which I'm learning about derived functors also does not require additivity of the functor or abelian categories. It has been changed.