symmetric monoidal (∞,1)-category of spectra
A right adjoint to restriction of scalars. The dual notion to extension of scalars .
Here is one special case.
For $R$ a ring, write $R$Mod for its category of modules. Write Ab = $\mathbb{Z}$Mod for the category of abelian groups.
Write $U\colon R Mod \to Ab$ for the forgetful functor that forgets the $R$-module structure on a module $N$ and just remembers the underlying abelian group $U(N)$.
The functor $U\colon R Mod \to Ab$ has a right adjoint
given by sending an abelian group $A$ to the abelian group
equipped with the $R$-module struture by which for $r \in R$ an element $(U(R) \stackrel{f}{\to} A) \in U(R_*(A))$ is sent to the element $r f$ given by
This is called the coextension of scalars along the ring homomorphism $\mathbb{Z} \to R$.
The unit of the $(U \dashv R_*)$ adjunction
is the $R$-module homomorphism
given on $n \in N$ by