This article definitely assumes more knowledge than basic abstract algebra - there are also some key facts used from set theory assumed (i.e. In a proof, it is assumed to be known that if a compositiom of two maps is the identity map on one set, and the reverse composition is the identity on the other set, then the map is a bijection - it also assumes knowledge from abstract algebra that the composition of two homomorphisms is a homomorphism).
I have only looked in the first chapter, so I cannot speak further on that atm. That said, I appreciate that a text talks about the universal property - I did not encounter a definition of it in a text when in grad school. I only encountered it because lecturers at my grad program made sure to talk about it.
> In a proof, it is assumed to be known that if a compositiom of two maps is the identity map on one set, and the reverse composition is the identity on the other set,
This is something math and computer science students typically learn in the first two weeks in their mandatory math lectures at least at German universities
I mean: It really makes sense to put it at the beginning since otherwise it's ugly to understand why a diffeomorphism is defined as it is:
You surely know that an isomorphism of sets (bijective function) has an inverse that is also an isomorphism of sets.
For differentiable functions a similar statement does not hold in general (just consider [-1,1] -> [-1,1]; x \mapsto x^3; its inverse is not differentiable everywhere on [-1,1]; so its inverse exists as an isomorphism of sets, but not as a differentiable function). Since diffeomorphisms for R^d are introduced in the 2nd semester for math students (typically in the context of the inverse function theorem), one better has already understood the basics before.
I'm in the middle of writing https://arbital.com/p/universal_property_outline/ which is precisely an Intro to the Universal Property - comments, feedback and assistance welcome, if you care to help :)
I have only looked in the first chapter, so I cannot speak further on that atm. That said, I appreciate that a text talks about the universal property - I did not encounter a definition of it in a text when in grad school. I only encountered it because lecturers at my grad program made sure to talk about it.