Categorical duals for finitely-generated projective modules

Question

For a not necessarily commutative ring $R$, its category of bimodules $_BMod_B$ has a monoidal structure given by $\otimes_R$. Consider now an object $M$ in the subcategory whose objects are finitely-generated projective as left modules. Will $Hom(M,R)$, the bimodule of left $R$-module maps be a left dual for $M$ in$_BMod_B$, ?

Answer 1

According to Etingof et al:

[Exercise 1.10.16] Let $A$ be an algebra. Show that $M \in A$-bimod has a left (respectively,right) dual if and only if it is finitely generated projective when considered as a left (respectively,right) A-module.

So it seems a dual basis argument settles the question.