Hom-tensor adjunctions

Let $A$ be a ring (which might or might not be commutative), and let $M,N$ and $K$ be three bi-modules over $A$.

There are two hom-tensor adjunctions. One says that

$Hom_A(M\otimes_A N, K) \cong Hom_A(M,Hom_A(N,K))$.

The other says that

$Hom_A(M\otimes_A N, K) \cong Hom_A(N,Hom_A(M,K))$.

Are these isomorphisms of bimodules?

If so, does this mean that the two bimodules $Hom_A(N,Hom_A(M,K))$ and $Hom_A(M,Hom_A(N,K))$ are isomorphic?

Be careful. It's cleanest to describe the tensor-hom adjunction with three different rings instead of one, to make it as hard as possible to accidentally write down the wrong thing, so let $A, B, C$ be three different rings, let $_A M_B$ be an $(A, B)$-bimodule, let $_B N_C$ be a $(B, C)$-bimodule, and let $_A K_C$ be an $(A, C)$-bimodule. Then

$$\text{Hom}_C(M \otimes_B N, K) \cong \text{Hom}_B(M, \text{Hom}_C(N, K))$$

as $(A, A)$-bimodules, and

$$\text{Hom}_A(M \otimes_B N, K) \cong \text{Hom}_B(N, \text{Hom}_A(M, K))$$

as $(C, C)$-bimodules.

Specializing to the case that $A = B = C$ shows that your notation is sloppy (to be fair, so is mine): when you write $\text{Hom}_A$ you haven't been careful about whether this means left $A$-module or right $A$-module homomorphisms, and it has different meanings in the different parts of your adjunctions unless $A$ is commutative and $M, N, K$ are plain $A$-modules, in which case there's no need to make left/right distinctions.

(Specifically, $\text{Hom}_A$ means left the second, fifth, and sixth times you used it, but right the first, third, and fourth times.)

It turns out the two bimodules you mention are isomorphic. Adjunction in general gives you the bijection you described. However, in the proof of the Hom/Tensor adjunction, the map that you define for the bijection can be seen to also be a homomorphism. Really you have to write out the proof in detail, and observe that you are dealing with homomorphisms. More information can be found here:

Adjointness of Hom and Tensor

In fact the exact statement you are asking about is mentioned here:

http://en.wikipedia.org/wiki/Tensor-hom_adjunction