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Aug 26 at 21:47 comment added Stahl @YiningChen I resolved this issue for myself by deciding to work with general $\infty$-categories, rather than trying to work in the dg-setting, so I never did get a definitive answer to this question. However, a new paper was posted to arXiv related to this topic, titled "k-linear Morita theory," by Matteo Doni. I haven't read it in detail, so I don't know whether it answers this question, but at a glance it might be possible that the equivalence detailed therein is symmetric monoidal.
Aug 26 at 9:30 comment added Yining Chen Hello, do you solve this problem? I have the same question.
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Mar 4, 2021 at 4:31 history edited Stahl CC BY-SA 4.0
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Mar 4, 2021 at 4:09 history asked Stahl CC BY-SA 4.0