The change-of-monoid adjunction between categories of modules induced by a morphism of monoids

Question

Let $\mathcal{M}$ be a cocomplete closed symmetric monoidal category. Let $A, B$ be monoids in $\mathcal{M}$ and $f: A \rightarrow B$ be a morphism of monoids. The morphism $f$ induces the extension of scalars functor and the restriction of scalars functor between the categories ${_A\mathrm{Mod}}$ of left $A$-modules and ${_B\mathrm{Mod}}$ of left $B$-modules. These functors form an adjunction between these categories. I am looking for a reference for this adjunction.

In other words, I am looking for a reference to the change of rings adjunction but in the more general case of monoids in a cocomplete closed symmetric monoidal category.

Answer 1

Certainly overkill, but this result is a special case of `change of operads' since the given morphism of monoids induces a morphism from the operad $O$ for left $A$-modules to the operad $P$ for left $B$-modules. A published reference is Corollary 5.1.1 in Homotopical Adjoint Lifting Theorem. Tracing through the proof and ignoring all the stuff about model categories, the key categorical result we rely on there is the Adjoint Lifting Theorem, for which we reference Borceux's volume 2, Theorem 4.5.6 on page 226. When we use this result, we take (using the notation of diagram 3.3.6 in our paper) $L = R = Id$ on the underlying category, and so $f^*$ is given by restriction along $f$. In case you don't want to rely on Borceux for the left adjoint, $f_!$ you can also get a direct construction from Section 4 of Resolution of coloured operads and rectification of homotopy algebras paper by Berger and Moerdijk, which again works in the generality of operads.