The question is as shows:
Let $[A,B]$ be a bimatrix game such that both A and B are diagonal matrices with nonnegative diagonal entries. Show that any diagonal matrix $(p_{ij})$ such that $(p_{ij}) \geq \Sigma_{i,j}p_{ij}=1$, is a CE. (Correlated Equilibrium)
Any insight on how to solve this be helpful!
If the elements off the diagonal are zero then there are no profitable deviations when the other player follows the correlated equilibrium:
Suppose player 2 follows the correlated equilibrium. Then you must prove that no other strategy will improve your payoff in expected value. Showing this is very simple in your case: suppose you get the $i$th signal. Player 2 will play his $i$th action and your best response is also to play your $i$th action since the payoff matrix is diagonal. Use this argument for each signal to see that the correlated equilibrium maximizes your payoff for each signal $i$, so it must maximize your expected payoff as well, that is, $\sum p_{I}\pi_{I}$.