I have shown that a language $A$ is NP-complete $\iff$ its complement $\overline{A}$ is co-NP-complete.
Also, I have shown that CNF-SAT is NP-complete.
Since UNSAT is the complement of SAT, then CNF-UNSAT is the complement of CNF-SAT and CNF-UNSAT is co-NP-complete, right?