Let $\mathcal A$ be an abelian category. In [this lecture](https://youtu.be/7RNh_cxDYmk), Thomas Nikolaus


1. Defines the unbounded derived category $\mathcal D(\mathcal A)$ as $\mathcal K(\mathcal A)[W^{-1}]$, where $\mathcal K(\mathcal A)=N\_{\mathrm{dg}}(\operatorname{Ch}(\mathcal A))$ and $W$ is the set of quasi-isomorphisms of chain complexes; and
2. Defines the notion of left-derived functor from an $\infty$-category with a set of weak equivalences to an arbitrary $\infty$-category, and shows that the left derived functor of a functor from $\mathcal K(\mathcal A)$ to an arbitrary $\infty$-category can be computed as a limit, provided it exists.


Point (1) is identical to the construction of the unbounded derived category of an abelian category as a triangulated category, before you know anything about K-injective complexes (Lurie’s approach in Higher Algebra is to require the existence of K-injectives to define the $\infty$-categorical unbounded derived category).
Point (2) is identical to Deligne’s approach to defining derived functors ([05S7](https://stacks.math.columbia.edu/tag/05S7)).


My question is, where is (2), or more generally an $\infty$-categorical notion of derived functor, written down? I wasn’t even able to find in HTT, HA, or SAG a definition of derived functor from an $\infty$-category with weak equivalences to another $\infty$-category, but I may have overlooked it in the thousands of pages. I’m especially interested in Nikolaus’s computation of derived functors out of $\mathcal K(\mathcal A)$ as a limit/colimit because it’s easy to connect to everything that’s written down in the language of triangulated categories (e.g. you immediately know you can compute the derived tensor product using a K-flat resolution etc.).



Derived functors out of an unbounded derived $\infty$-category
An account of derived functors between ∞-categories equipped with weak equivalences and fibrations can be found in Section 7.5 of Cisinski's *Higher Categories and Homotopical Algebra*. This setting is sufficient to treat the case of the unbounded derived category of an abelian category.


A more general treatment of derived functors in the setting of ∞-categories equipped with a calculus of fractions can be found in Section 7.2. See, in particular, Theorem 7.2.8 and Corollary 7.2.9, as well as Remark 7.2.21, which connects this setting to the setting of fibrations.


