Look, it's 2019. "Setting up category-theoretic language" is something that should begin in the first lesson of the first course in formal, non-K12 math, starting with the use of ETCS or a comparable "structural" set theory as the 'naïve' foundation of choice. It provides countless advantages not just in abstract algebra, but in a wide variety of other subfields of math as well.
To be clear, how much category theory are you envisioning setting up in a first course on abstract algebra? Are you basically just talking about a modern treatment of morphisms in the abstract sense, or do you also mean discussing things like functors?
My perspective on this is basically informed by the following: I like category theory a lot. I've chatted with Baez about this on a few occasions and insofar as I need to use algebra, I like the category theoretic formalisms that come along with it. I think if you had students with a lot of mathematical maturity but no prior exposure to abstract algebra (except maybe a rigorous treatment of linear algebra), you could accomplish what you're proposing.
To your point, one of my favorite math textbooks is Aluffi's Chapter 0. You're probably familiar with it, but if not: it builds up abstract algebra in a rigorous and modern fashion alongside category theory. However, there are a few caveats here:
1. Aluffi explicitly states it's a textbook for upper level undergrads and preferably graduates. In my experience, when authors say this they actually mean it's appropriate as a year one graduate course. That's not the time to learn abstract algebra for the first time!
2. Aluffi does a great job of covering things like morphisms and categories, but the size of the book is massive. He doesn't have any nontrivial coverage of deeper category theoretic concepts like functors until much later in the book; it's probably far enough in that you couldn't reasonably cover it in a single semester course.
3. I really don't think most math majors would benefit from it. Those who are applied math majors will get questionable benefit from a slower, more comprehensive approach to algebra than a faster approach that lets them get to applications. Moreover, not all pure math majors have the mathematical maturity to approach category theory before grad school. Proofs in traditional abstract algebra are a lot less abstract than category theory, and it's easier to motivate them even if they aren't ultimately as elegant.
The problem with "a faster approach that lets [you] get to applications" is that it will depend a lot on what applications you have in mind. And often it's not even faster in any real sense - there's a whole lot of pointless duplication involved in tailoring things to a lower level of abstraction. It may be better to begin with a more effective explanation of what sorts of "mathematical maturity" we're actually seeking here, so that attaining it is easier in the first place for the average student. For all their supposed "unintuitiveness", category-theory-based explanations do this quite well, in a way that I haven't really seen elsewhere - I mean, this very thread is one where we're discussing software developers learning category theory and benefiting from it, so why couldn't this be also a part of mathematical training?
I strongly disagree. From experience teaching category theory to undergraduates, it is totally obtuse before that have a large pool of examples on which to draw. In a very first course, students are still struggling to learn concepts from linear algebra, which will be the bread and butter of the rest of their mathematics. In a second course, they are struggling to learn why the definition of "linear map" is different from the definition of "matrix" they learned in their first course. You could say the same thing about metric spaces or topology.
Perhaps the third course is a nice place to start introducing some category theory, since they will now have actual examples on which to draw, and learn about the unifying concepts of "morphism", "categorical product", and so on. They might even be ready for some interesting functors, such as the fundamental group of a topological space.
If you were to teach category theory as a very first course, what would you teach? What examples of categories do you have? What examples of functors?
