Proposition 19.1.7. Let \(C\) be a symmetric monoidal \(\infty \)-category. Then the following commutative square is a pullback square:
In particular, passing to vertical fibers over \(R \in \CAlg (C)\) induces an equivalence \[ \Mod _R(C) \iso \LMod _{R'}(C), \] where we denote the underlying associative algebra of \(R\) by \(R'\).
Proof. This follows by combining Glasman (2014), Proposition 7 with [Lurie (2017), Proposition 4.5.1.4]. More explicitly, the vertical functors are cartesian fibrations, the top comparison preserves cartesian morphisms because it does not change underlying objects, and the cited results identify it as an equivalence on every vertical fiber. Hence the square is a pullback. □
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