Proposition 19.2.11. Let \(C\) and \(D\) be monoidal \(\infty \)-categories which admit geometric realizations and whose tensor products preserve geometric realizations separately in both variables. If \(F\colon C\to D\) is a monoidal functor which preserves geometric realizations, then for every associative algebra \(B\in \Alg (C)\), right \(B\)-module \(M\), and left \(B\)-module \(N\), there is a natural isomorphism \[ F(M)\otimes _{F(B)}F(N)\iso F(M\otimes _BN). \]
Proof. The monoidal structure on \(F\), together with the functorial construction of the coherent bar diagram from Remark 19.2.10, gives an isomorphism of simplicial objects \[ \Bar _{F(B)}(F(M),F(N))_{\bullet }\iso F(\Bar _B(M,N)_{\bullet }). \] Taking geometric realizations and using that \(F\) preserves them gives the asserted isomorphism. □
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