Let $M$ be an $A,B$-bimodule and $C$ a ring. Show that the functor $\text{Hom}_A(-,M)$, from the opposite category of $A,C$-bimodules to $C,B$-bimodules, is a right adjoint. [Hint: its left adjoint is essentially $\text{Hom}_B(-,M)$, with "op" in the right places.]
Following the hint, I tried to prove that $\text{Hom}_{C,B}( \text{Hom}_B(N,M),L) \cong \text{Hom}_{A,C}(N, \text{Hom}_A(L,M))$ where $N$ is an $C,B$-bimodule and $L$ is an $A,C$-bimodule. However, I am stuck on showing there there is a bijection between the two sets, much less showing that the bijection is natural. The problem is that any $C,B$-bimoule homomorphism that takes an (right $B$-module homomorphism) $g$ to an element $l \in L$ may not be surjective, so that it's unclear to me how to define a homomorhpism in $\text{Hom}_{A,C}(N, \text{Hom}_A(L,M))$ that would naturally correspond to a map in $\text{Hom}_{C,B}( \text{Hom}_B(N,M),L)$. Also, I am not sure what the hint " "op" in the right places" mean.
EDIT: after seeing this post, it remains to figure out what exactly the hint means by " "op" in the right places" and how it's justified to take the Hom on the right in the isomorphism above rather than on the left. Can anyone provide clarification on this?
If $F:\mathcal{C}\to \mathcal{D}$ is a functor, its left/right adjoint goes from $\mathcal{D}$ to $\mathcal{C}$.
Applying to your case, you have $F=\text{Hom}_A(-,M):\mathcal{C}^{op}\to \mathcal{D}$ which tells you that you have to consider its adjoint as a functor $\mathcal{D}\to \mathcal{C}^{op}$ (here $\mathcal{C}$ and $\mathcal{D}$ denote the categories of modules you mention).