Is it possible to formulate category theory without set theory?

I posted this question over at math stackexchange after the thread about category theory and haskell from yesterday[1].

I found Carl Mummert's answer especially illuminating and definitely recommend reading it. He compares category theory to a strongly typed language, and set theory to a weakly typed one.

[1] https://news.ycombinator.com/item?id=10527428