I learned the tensor-hom adjunction months ago, from several different references, like in a standard Homological algebra textbook(Rotman’s Introduction to Homological Algebra) or from a standard category theory textbook(Riehl’s Category Theory in Context)
The adjunction of two functors $F: \mathcal{C} \to \mathcal{D}$, only says that the two morphism sets $Mor(Fc, d) \cong Mor(c, Fd)$, i.e. there is an bijection of two sets. But the tensor-hom adjunction has something more. It’s an isomorphism between two modules, quoted as following from wekipedia:
Say $R$ and $S$ are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules):
$\mathcal{C}=\operatorname{Mod}_S \quad$ and $\quad \mathcal{D}=\operatorname{Mod}_R$.
Fix an $(R, S)$-bimodule $X$ and define functors $F: D \rightarrow C$ and $G: C \rightarrow D$ as follows:
$$
\begin{aligned}
&F(Y)=Y \otimes_R X \quad \text { for } Y \in \mathcal{D} \\
&G(Z)=\operatorname{Hom}_S(X, Z) \quad \text { for } Z \in \mathcal{C}
\end{aligned}
$$
Then $F$ is left adjoint to $G$. This means there is a natural isomorphism
$\operatorname{Hom}_S(Y \otimes_R X, Z) \cong \operatorname{Hom}_R\left(Y, \operatorname{Hom}_S(X, Z)\right)$.
This is actually an isomorphism of abelian groups. More precisely, if $Y$ is an $(A, R)$
bimodule and $Z$ is a $(B, S)$ bimodule, then this is an isomorphism of $(B, A)$ bimodules.
This is one of the motivating examples of the structure in a closed bicategory.
The argument “This is actually an isomorphism of $(B, A)$ bimodules”, seems beyond the standard statement of being an adjunction. I can only check it manually.
How does this hold from the categorical view point? What extra categorical properties is necessary for this? I believe that I lack some basic knowledge about this. I checked the other two questions: Adjunction in Abelian Categories and $\mathrm{Hom}$-sets in categories with finite biproducts have the structure of a commutative monoid: a reference for a proof but still cannot figure it out.
The extra properties should prove that addition and actions of B and A are all preserved. The linked questions seem can be used to prove that addition can be preserved, but how about the multiplications?
Sorry for kind of being fuzzy. This question has perplexed me for a while. Any answer or comment or reference or any kind of helps would be appreciated. Thank you.
Best Answer
To prove that the adjunction automatically respects the abelian group structures it suffices to know that the abelian group structure on the homsets in a category of modules is determined by the behavior of biproducts in that category, and then observe that both the functors $\text{Hom}(X \otimes Y, Z)$ and $\text{Hom}(Y, \text{Hom}(X, Z))$ preserve biproducts (in all three variables, but it would suffice to pick one). This is discussed in the questions you linked to, the second one of which links to my blog post A meditation on semiadditive categories.
To prove that the adjunction automatically respects the $(B, A)$-bimodule structures on both sides it suffices to use functoriality: if $Y$ is an $(A, R)$-bimodule then it is a right $R$-module on which $A$ acts, and by functoriality this action of $A$ induces a right action on $\text{Hom}(X \otimes Y, Z)$ and on $\text{Hom}(Y, \text{Hom}(X, Z))$. Similarly for the action of $B$ on $Z$. And a natural isomorphism respects functoriality by definition.