Let $G$ be an arbitrary directed graph. Does $G$ always have the same strongly connected components on $G$ as on $G^*$? Here, $G^*$ is the inverted graph of $G$ (i.e., $(u,v)\in E \rightarrow (v,u) \in E^*$).
I'm thinking yes because otherwise the algorithm of Kosaraju would not work at all? But do not have another explanation for this.