TITLE: Derived functors are Kan extensions
QUESTION [8 upvotes]: In this short paper by G. Maltsiniotis derived functors are presented as Kan extensions along the localization functor. 
I began studying derived categories only a couple of months ago, so I'm not at all an expert, but this approach is completely new to me, as I've not found it in any of the book in my hand (the most advanced is Categories and Sheaves but I've never read it sistematically, so maybe I haven't noticed it there).
Can you give, if it exists, a reference which rewrites some of the basics on derived functors taking into account the "Kan extensions perspective"?

REPLY [3 votes]: I cannot comment so I post this as an answer: derived functors are kan extensions via definition 13.3.1, 7.3.1 and 2.3.2 of Kashiwara and Shapira "Categories and Sheaves" ( well, they do not use the words kan extension in 2.3.2).