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

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Why does the following construction describe the Serre functor?

In the book "Spectral Agebraic Geometry" that Jacob Lurie is currently writing, he gives a construction (11.1.5.1), which describes the Serre functor: it has already been shown that any proper $A$-lin …