This problem occured to me, when trying to find a Morita invariant for finite dimensional algebras.

Suppose $\Lambda$ and $\Gamma$ are two self-injective $k$-algebras ($k$ being a field) which are Morita equivalent. The Morita equivalence should be given by the $\Lambda-\Gamma$-bimodule $P$ and the $\Gamma$-$\Lambda$-bimodule $Q$. Each of the algebras has a Morita self-equivalence given by the bimodules $D\Lambda$ and $D\Gamma$, where $D$ is the standard $k$-duality.

What can one say about the commutativity of the two compositions, i.e. is $$D\Gamma\otimes_{\Gamma} > Q\otimes_\Lambda - \cong > Q\otimes_\Lambda D\Lambda > \otimes_\Lambda - $$ in general?

Further comment: The Morita self-equivalence can equivalently be given as $D Hom_\Lambda(-,\Lambda)$. This is called the Nakayama functor.

I did not succeed in finding a counterexample nor giving a proof and I don't know what to expect.