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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

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Is the functor $mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C})$ cohomologically full and fai...

This seems to me to be unproblematic (if messy to prove). Try if this does the job: Proposition 12.15(2) in Bespalov-Lyubashenko-Manzyuk, Pretriangulated A_infty-categories
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