Give a Search Problem in co-NP
Specifically speaking, 3-SAT is an example of NP-Complete problem. But your problem is only considered with NP problems.

We have the decision problem:


  Is a SAT problem in NP?


Obviously, it is. Because you could always verify a "certificate" in time linear to your input size (values of your literals, that is a sequences of 0s and 1s).

For your question:


  Is a un-SAT problem in co-NP? (complement to SAT problem)


We could find the following definition:


  A decision problem X is a member of co-NP if and only if its
  complement X is in the complexity class NP [1].


Therefore, un-SAT is co-NP by definition.