Proposition 19.6.5 (Monadicity of module categories). Let \(C\) be a monoidal \(\infty \)-category, let \(D\) be left-tensored over \(C\), and let \(A \in \Alg (C)\). The forgetful functor \[ U_A\colon \LMod _A(D) \longrightarrow D \] is monadic, and its associated monad is \(A \otimes -\colon D \to D\), with unit and multiplication induced by those of \(A\).
Proof. The left adjoint exists by Proposition 19.1.13, and its composite with \(U_A\) is \(A \otimes -\). The forgetful functor is conservative by [Lurie (2017), Corollary 4.2.3.2]. Moreover, every \(U_A\)-split simplicial object admits a geometric realization which is preserved by \(U_A\); this is [Lurie (2017), Lemma 4.7.3.12]. The claim follows from Theorem 19.6.3. □
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