Is there an adjoint functor to the hom bifunctor in the category of $R$-modules, $R$ is commutative unital ring. I've seen that for $W\in R-\mathrm{\!Mod}$ ,both $\mathrm{Hom}(W,-)$ and $\mathrm{Hom}(-,W)$  have adjoint functors.
adjoint of Hom(-,W)
this leaves me with one question:
$C:= R-\mathrm{\!Mod}$
does $\mathrm{Hom}(-,-):C^{op}\times C \to C$ have an adjoint functor?
 A: A functor between additive categories which has either a left or a right adjoint must be additive, but $\text{Hom}$ is not; it is bilinear rather than linear. The tensor bifunctor also fails to have an adjoint for the same reason.
A: I found a counterexample:
$C:= \mathrm{\mathbb{Z}-Mod}, D:=C^{op}\times C$
$R(-):=Hom(-,-) : D\to C$ , $L$ is the left adjoint of $R$.
Then $Hom_D(L(-),-)\cong Hom_C(-,R(-)):C^{op}\times D\to Set$
$\mathbb{Z}_i := \mathbb{Z}/i\mathbb{Z}$
$Hom_D(L(\mathbb{Z}_r),(\mathbb{Z}_n,\mathbb{Z}_m))\cong Hom_C(\mathbb{Z}_r,R(\mathbb{Z}_n,\mathbb{Z}_m))$
Suppose $L(\mathbb{Z}_r)=(M,N) \in D$
$Hom_D((M,N),(\mathbb{Z}_n,\mathbb{Z}_m))\cong Hom_{C^{op}}(M,\mathbb{Z}_n)\times Hom_D(N,\mathbb{Z}_m)\cong Hom_{C}(\mathbb{Z}_n,M)\times Hom_D(N,\mathbb{Z}_m)$
$|Hom_C(\mathbb{Z}_r,Hom(\mathbb{Z}_n,\mathbb{Z}_m)|=gcd(r,n,m)$
$|Hom_{C}(\mathbb{Z}_n,M)|\times |Hom_D(N,\mathbb{Z}_m)|=gcd(r,n,m)$
take n, we get $|Hom_D(N,\mathbb{Z}_m)|$ must be $1$.
by the same way $|Hom_{C}(\mathbb{Z}_n,M)|=1$
we have $gcd(r,n,m)=1$ for all n,m , that leads contradiction.
(I'm not sure whether I made a mistake or not)
