Question regarding an answer on dual of a dual cone

Question

Regarding this answer. We can take $K=\{(0,0)\}\cup\{(a,b):~a>0,b>0\}$. We have $K^*=\{(a,b):~a\geq 0,b\geq 0\}$. Let us take $C=\{(0,0)\}\cup\{(a,b):~a\geq 0,b>0\}$ and notice $K \subset C$. But we do not have $K^{**}\subset C$ as stated in your answer, because $K^{**}=K^*$ that is larger than $C$.