  This is the definition one finds in most textbooks of category theory. As such it explicitly relies on a set theoretical background and language. An alternative, suggested by Lawvere in the early sixties, is to develop an adequate language and background framework for a category of categories. We will not present the formal framework here, for it would take us too far from our main concern, but the basic idea is to define what are called weak \(n\)-categories (and weak \(\omega\)-categories), and what had been called categories would then be called weak 1-categories (and sets would be weak 0-categories). (See, for instance, Baez 1997, Makkai 1998, Leinster 2004, Baez &amp; May 2010, Simpson 2011.)