Tensor–Hom adjunction lemma

Natural isomorphism between Hom out of a tensor product and Hom into a Hom-module.

Tensor–Hom adjunction lemma

Tensor–Hom adjunction lemma: Let $R,S$ be unital rings , let ${}_S M_R$ be an $(S,R)$ -bimodule , let ${}_R N$ be a left $R$ -module , and let ${}_S P$ be a left $S$ -module. Then there is a natural isomorphism of abelian groups

\operatorname{Hom}_S(M\otimes_R N,\,P)\;\cong\;\operatorname{Hom}_R\!\bigl(N,\,\operatorname{Hom}_S(M,P)\bigr),

functorial in $N$ and $P$ .

This is the concrete form of the Tensor–Hom adjunction for a bimodule , relating tensor products and Hom-modules .