Theory of Complexity - Relation between a class and its complement
The first part is easy since there is a $C$-complete problem that belongs to $coC$ and since all problems in $C$ are reduced to that complete problem, $C \subseteq coC$

As for the second part let $L_C$ be a $C$-complete problem that belongs to $coC$. Then by definition of $coC$, $\overline{L}_C \in C$. So, $\overline{L}_C$ are in $C$ (decidable in $C$). Also since $L_C$ is $C$-complete, for any language $L \in C$ there is a reduction $f$ such as $x \in L \Leftrightarrow f(x) \in L_C$.

Now, if a problem $L \in coC$ then $\overline{L} \in C$ and $x \in \overline{L} \Leftrightarrow f(x) \in L_C$ meaning that $x \in L \Leftrightarrow f(x) \in \overline{L}_C$. So $L$ is reduced to $\overline{L}_C$ which is in $C$. Thus $coC \subseteq C$.