Tensor-Hom Adjunction In Monoidal Categories? Is there a generalization of the tensor-hom adjunction to monoidal categories, or is it a special property of $\mathsf{Mod}$-$R$?


If $\mathcal{C}$ is closed symmentric monoidal category, and $\mathbb{T}$ is symmetric monoidal monad on $\mathcal{C}$, then the category of $\mathbb{T}$-algebras caries a structure of closed symmetric monoidal category.

this result is well known, and was proved be Anders Kock in the '70.
For the definition of symmetric monoidal monad, you can look at this paper.
In our case the monad is of free R-modules.
(I don't have enough points to elaborate...)
