Let $A$ be a commutative ring with a unit element. Let $M$ and $N$ be $A$-modules. Let $M^v$ and $N^v$ be the dual modules. In general, do we have $M^v \otimes N^v \cong (M\otimes N)^v$? It is definitely true when M and N are free. I believe (though haven't worked out the details) that it is true when M and N are projective. Both Atiyah & Macdonald and Lang don't say anything on the matter.

I came up with this question while studying $Pic(A)$: defined as the group of isomorphism classes of projective modules of rank 1. If I and J are projective $A$-modules of rank 1, then the fact that $Pic(A)$ is a group immediately implies $I^v \otimes J^v \cong (I\otimes J)^v$, as both are the inverse of $I\otimes J$.