Tensor left adjoint as a bi-functor?

Question

Working in the category of modules over a fixed ring $A$ $\operatorname{Mod}_A$, we have the adjunction: $$F(X) = X\otimes_A M \dashv G(X) = \operatorname{Hom}_A(M,X)$$ for a fixed module $M$. Is there a version where we do not fix $M$. That is, is there a right adjoint to the following functor: $$H: \operatorname{Mod}_A\times \operatorname{Mod}_A \to \operatorname{Mod}_A: H(X,Y) = X\otimes_A Y.$$ I am also interested in a version where the ring is not fixed either. I am not sure what category to work from in this case though.

Answer 1

Multivariable adjunctions do not work like that. For instance, if $(X, Y) \mapsto X \otimes_A Y$ had a right adjoint, then it would have to preserve direct sums, i.e. the comparison $$(X_0 \otimes_A Y_0) \oplus (X_1 \otimes_A Y_1) \to (X_0 \oplus X_1) \otimes_A (Y_0 \oplus Y_1)$$ would have to be an isomorphism. But, in fact, $$(X_0 \oplus X_1) \otimes_A (Y_0 \oplus Y_1) \cong (X_0 \otimes_A Y_0) \oplus (X_0 \otimes_A Y_1) \oplus (X_1 \otimes_A Y_0) \oplus (X_1 \otimes_A Y_1)$$ so $(X, Y) \mapsto X \otimes_A Y$ cannot have a right adjoint (assuming $A \ncong \{ 0 \}$).