From module to bimodule

Observation 19.2.4. There is an operad map \(\oBMod \to \oMod \) which sends \(\fm \) to \(\fm \) and sends both \(\fa _-\) and \(\fa _+\) to \(\fa \). Composing it with the inclusion \(\oLMod \hookrightarrow \oBMod \) gives the canonical map \(\oLMod \to \oMod \), and similarly for \(\oRMod \). It follows that for a commutative algebra \(A\) in a symmetric monoidal \(\infty \)-category \(C\), restriction along these operad maps defines forgetful functors

Here the left diagonal equivalence is the one from Proposition 19.1.7. The right one follows by applying Observation 19.2.5, since a commutative algebra is canonically isomorphic to its opposite algebra.