TITLE: Dual of a bimodule
QUESTION [9 upvotes]: For a noncommutative ring $R$, and an $R$-$R$-bimodule $B$, is there a "correct/natural" notion of a dual bimodule? I am interested, really, when $B$ is projective as a left $R$-module.
Note: Switched from Stackexchange, since no answers

REPLY [13 votes]: As explained in more detail in this blog post linked by Jakob in the comments, every $(A, B)$-bimodule $M$ has two natural duals:

If $M$ is finitely generated projective as a left $A$-module, it has a left dual given by the $(B, A)$-bimodule $\text{Hom}_A(M, A)$.
If $M$ is finitely generated projective as a right $B$-module, it has a right dual given by the $(B, A)$-bimodule $\text{Hom}_B(M, B)$.

These duals come from thinking of an $(A, B)$-bimodule as a 1-morphism in the Morita 2-category whose 

objects are rings
1-morphisms are bimodules
2-morphisms are bimodule homomorphisms

and applying the general equational definition of dual or adjoint 1-morphisms in a 2-category given by the zigzag identities (the one which, applied to the 2-category of categories, produces left and right adjoints).