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.