tensor functor as left adjoint

Let \(R\) be a ring, \(M\) an \(R\)-module and \(\mathrm{RMod}\) the category RMod. Then the tensor functor

\begin{align*} (M \otimes_{R} -): \mathrm{RMod} \rightarrow \mathrm{Ab} \end{align*}

is a left adjoint

Consider for arbitrary \(N,L \in \mathrm{Ob}(\mathrm{RMod})\) the maps

\begin{align*} \varphi: \mathrm{Hom}_{\mathrm{RMod}}((M \otimes_R N), L) \rightarrow& \mathrm{Hom}_{\mathrm{RMod}}(M, \mathrm{Hom}_{\mathrm{RMod}}(N,L)) \\ (f: M \otimes_R N \rightarrow L) \mapsto& (f': m \mapsto (n \mapsto f(m,n)) \\ \varphi^{-1}: \mathrm{Hom}_{\mathrm{RMod}}(M, \mathrm{Hom}_{\mathrm{RMod}}(N,L)) \rightarrow& \mathrm{Hom}_{\mathrm{RMod}}((M \otimes_R N), L) \\ (g: M \rightarrow (h: N \rightarrow L)) \mapsto& g': (m,n) \mapsto (g(m))(n) \end{align*}

see:

• module homomorphism to the hom object induced bilinear map
• tensor product and fixing component of a homomorphism