What is the precise definition of the prefix "co" in mathematics?

Question

Given a notion "A" in mathematics, in many cases "coA" is also defined. Here are some common examples:

1. sine and cosine;
2. tangent and cotangent;
3. secant and cosecant;
4. function and cofunction;
5. morphism and comorphism;
6. functor and cofunctor;
7. domain and codomain;
8. limit and colimit;
9. set and coset;
10. product and coproduct;
11. fibration and cofibration;
12. homology and cohomology;
13. homotopy and cohomotopy;
14. prime and coprime;
15. vector and covector;

and the list goes on. My question is, what is the generally accepted meaning of the prefix "co"? Given a mathematical notion "A", when is "coA" also defined? Also, is "cocoA" always the same as "A"? (Here I am only asking about mathematical terminologies, so "coconut" does not count.)

Edit: Among all the examples listed above, the pair puzzles me the most is "set and coset". A coset is defined in the context of a subgroup of a group. I am wondering if there is any reason to call it a coset.