I am delving a bit into category theory and something has me curious about opposite categories. I have checked several books and I can't seem to find an answer.
Given a category C, the opposite category is just the abstract category with the objects of C and with the arrows of C reversed. However, the opposite category can sometimes be realised (is equivalent to) a category where the objects are sets with additional structure, and the arrows are homomorphisms of these structures.
For instance one of the examples on Wikipedia is that the opposite category of commutative rings is equivalent to the category of affine schemes.
Question. How would one in general, given a category C, find a category where the objects are mathematical structures with underlying sets that satisfy additional axioms, and the morphisms are homomorphisms of those structures, and which is the opposite of the original category C?
For instance, if we take the category where the objects are groups, and the morphisms are group homomorphisms, what is its opposite category in the above sense? Is there some way to find this from the first-order axiomatization of groups?