Motivation:
Given any commutative ring $R$, the category $\mathrm{Mod}_R$ is both monoidal as $(\mathrm{Mod}_R,\otimes_{R},R)$ and as $(\mathrm{Mod}_R,\oplus,0)$, but the usual internal hom is right adjoint to the tensor product $\otimes_R$. However, according to the n-lab, an internal hom can be defined in any monoidal category as the right adjoint (if it exists) of the functor giving the monoidal structure, so if the direct sum $\oplus$ had a right adjoint, I'm guessing one could say that it is an internal hom for $(\mathrm{Mod}_R,\oplus,0)$, right? I think there is no such right adjoint, but is it the sole reason for not mentioning this possibility?
Question:
If a category can be given multiple monoidal structures, is it possible that more than one of them has a right adjoint? in other words, can a category have different monoidal closed structures? in case the answer is yes, is there an a priori reason to disregard some of them in favour of others as a notion of "usual" internal hom for the category? Are there cases where more than one monoidal closed structure is interesting?
Best Answer
Yes.
An example is the category of presheaves over a monoidal category. The category of presheaves for any (small) category is cartesian closed. If that small category has a monoidal structure, then we can define a monoidal structure on the category of presheaves via Day convolution. This category is monoidally closed because $\mathbf{Set}$ is complete. Neither of these structures will be "trivial" in general. In the case that the monoidal structure on the small category is the cartesian monoidal structure, then this turns out to produce the cartesian closed monoidal structure on the category of presheaves.
I'm not sure why we'd "disregard" structure. Usually one of the monoidal structures will be more relevant, but you'd certainly want to be aware of all structure that's available.