Corollary 2

Let A and B be Morita equivalent $k$-algebras, and let P be an invertible $(A,B)_k$-bimodule. Every $(B,A)_k$-bimodule inverse to P is isomorphic to the dual $P^*=$ Hom$_A(P,A)$ of P. More precisely, let Q be a $(B,A)_k$-bimodule that is an inverse of P, and let $\lambda : P\otimes_BQ\rightarrow_sA_d$ be an isomorphism of $(A,A)_k$-bimodules. There exists a unique mapping $\tau : Q\rightarrow P^*$ determined by the relation $\langle p, \tau (q)\rangle =\lambda (p\otimes q)$ for $p\in P$ and $q\in Q$, and $\tau$ is an isomorphism of $(B,A)_k$-bimodules.

The existence and uniqueness of the mapping $\tau$ are clear. It is a homomorphism of $(B,A)_k$-bimodules, and we have $\lambda = \Lambda \circ (1_P\otimes \tau )$. Since $\lambda$ and Λ are isomorphisms of $(A,A)_k$-bimodules (VIII, p. 101, Theorem 1), so is $1_P\otimes \tau$. By Lemma 1, the mapping $\tau$ is bijective.