I am reading this paper "Morita algebras", the link is here:http://ac.els-cdn.com/S0021869313001002/1-s2.0-S0021869313001002-main.pdf?_tid=7f6096b6-a581-11e6-9732-00000aab0f02&acdnat=1478588808_4e1053440cce588d704386a80a96f7bb.
At page 201, the remark below lemma 4.1 says that: "Let $B$ be a Frobenius algebra with $\nu_B$ not an inner automorphism. Then $B$ and $B_{\nu_B}$ are isomorphic as right $B$-modules. However, $B$ and $B_{\nu_B}$ are not isomorphic as $B$-bimodules because $\nu_B$ is not an inner automorphism" (Here $B_{\nu_B}=B$ as K-vector space, and $b_1 \cdot b=b_1 \nu_B(b)$ as right $B$-module )
$\textbf{My understanding and question:}$ We can get $B$ and $B_{\nu_B}$ are isomorphic as right $B$-modules by lemma 4.1. But I think the isomorphism $\varphi: B \rightarrow B_{\nu_{B}}$ is the form $\varphi(b)=\nu_B(b)$ for $b \in B$. So I can get they are not ismorphic as $B$-bimodules. I have not use the condition $\nu_B$ is not an inner automorphism. And I find that even $\nu_B$ is an inner automorphism, the map $\varphi$ is not a $B$-bimodule isomorphism. So I am confused here: Does the map $\varphi $ wrong? How to get $B$ and $B_{\nu_B}$ are not isomorphic as $B$-bimodules because $\nu_B$ is not an inner automorphism?