Reading around about the Nullstellensatz theorem, I saw that we can interpret the maps $V\to I(V)$ (where $V$ is an algebraic set) and $S\to Z(S)$ (where $S$ is an ideal), as functors which are adjoint, in some sense. I didn't find, however, more then side remarks about this. So I'm having a hard time trying to understand the full picture: what are the categories here, exactly? I can see how we can define the category of algebraic affine sets, but how do we think about the ideals as a category? Do we think of it as a subcategory of the modules over $k[\bar x]$? Looking at this question gives a way to do it, but why is it a good way- don't we want to save some of the algebraic structure?
Also, the radical ideals have some special property - the image of $I$ consists only of radical ideals. I wonder if there is a universal property for radical ideals, and whether it says something of value in the category of algebraic sets? Can it explain this special property?
I don't know a lot of category theory- just some basic definitions and constructions, so I am sorry if this make no sense at all, or if I am missing something trivial. Thanks in advance to anyone who can help in clarifying any of the points above, or maybe provide some references.