If problem is coNL-complete, is it NL-complete
If the language $A$ is $NL$-complete then 


$A \in NL$
Each problem in $NL$ is logspace reducible to $A$


Since $coNL = NL$ each problem in $coNL$ is logspace reducible to $A$ as well. Thus $A$ is $coNL$-complete.