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Jul 20, 2016 at 17:43 comment added Dylan Wilson See Thm 1.3.3.2 of Lurie's Higher Algebra. As for the other statement: derived functors themselves lift (example 1.3.3.4)... so just compose the lifts.
Jul 20, 2016 at 6:17 comment added mabli That is exactly the type of statement I was looking for. Thank you. Could you point me to a reference of the equivalence of functor categories you state in your answer. Also, if possible, a reference to your statement that "anything built up by composing derived functors will lift to the stable infinity categories".
Jul 19, 2016 at 20:26 vote accept mabli
Jul 19, 2016 at 19:57 history answered Jacob Lurie CC BY-SA 3.0