[sleepyams]

One interesting result which you may find interesting: https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem

I also highly recommend this survey paper by John Baez and Mike Stay: https://math.ucr.edu/home/baez/rosetta.pdf

There are plenty of interesting results in category theory, in fact your comparison to abstract algebra is apt. There is only so much you can say about an arbitrary group in general, or an arbitrary topological space, just like there is only so much you can say about an arbitrary category.