Is this a correct way to show that a problem is coNP-complete?
No, the steps are insufficient. You need to use a Karp reduction, since that's the type of reduction used in the definition of coNP-completeness. 

SAT is NP-complete, but probably not coNP-complete (unless NP=coNP); and coSAT is coNP-complete, but probably not NP-complete. This distinction would be lost if you used Cook reductions. 

Some questions on this site which are relevant are this one and that one, as well as a few others.