Corollary 19.1.14 ([Lurie (2017), Proposition 4.2.4.9]). Let \(C\) be a monoidal \(\infty \)-category and let \(A \in \Alg (C)\) be an algebra object such that the unit map \(e\colon \unit \to A\) is an isomorphism in \(C\). Then for every left \(C\)-tensored \(\infty \)-category \(D\) the forgetful functor \(\LMod _A(D) \to D\) is an equivalence of \(\infty \)-categories.
Proof. By Proposition 19.1.13, the forgetful functor admits a left adjoint \(F_A\colon D \to \LMod _A(D)\). Since the forgetful functor is conservative, it suffices to show that the unit map \(M_0 \to A \otimes M_0\) is an isomorphism. Since this map is given by tensoring the map \(e \colon \unit \to A\) by \(M_0\), this follows from the assumption on \(A\). Indeed, the triangle identity then identifies the underlying counit with the inverse of the unit at the underlying object, and conservativity implies that the counit is also an isomorphism. □
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