This question is a crosspost of the second part of this MSE question.

In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text Homotopy Limits Functors on Model Categories and Homotopical Categories, I learned a much more economic and conceptual definition as a Kan-extension along the localization functor.

Where can I find actual rigorous proof that in the abelian setting, the homologies of Kan extensions along localizations form universal $\delta$-functors?

In the MSE question, Zhen Lin proposed to simply calculate both in terms of acyclic resolutions, but I am looking for a proof using as little concrete calculations as possible, preferably employing only universal properties.

I'm a novice, so if you give a proof sketch, please be detailed.

Update: I know there are many great homotopy theorists here. That this question has remained unanswered but has not been downvoted makes me wonder - what's wrong with it?

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    $\begingroup$ I do not think it is possible to compare the two definitions in complete generality. I asked a related question some years ago – it is a fact that universal $\delta$-functors computed using resolutions annihilate injective objects, but I do not know if this is true for all universal $\delta$-functors. $\endgroup$ – Zhen Lin Jun 5 '15 at 12:31
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    $\begingroup$ @PhilippeGaucher Page 126, section 41.5. $\endgroup$ – Arrow Jun 12 '15 at 11:08
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    $\begingroup$ I am pretty sure that those definitions neither coincide nor are even closely related, unless in special cases. Note that there is extra structure you are missing on the right side: passage to homology after Kan extension. Defining an analogue of homology on homotopy category requires the notion of t-structure, and there can be very different (or even none) t-structures on the same homotopy category (look up "perverse sheaves"). The notion of delta-functor on its own is just not deep enough to claim some structural properties,it is just a statement of most basic properties of derived functors. $\endgroup$ – Anton Fetisov Jul 18 '15 at 8:38
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    $\begingroup$ However, in the specific important case of classical left and right derived functors, it is a theorem that they indeed form universal delta-functors. For proof see e.g. ([1], Chapter 2), in particular theorems 2.4.6 and 2.4.7. The fact that classical derived functors factor through Kan extension along localization w.r.t. quasi-isomorphisms is proved in ([1], Th. 10.5.6). [1]: C. A. Weibel, An introduction to homological algebra. $\endgroup$ – Anton Fetisov Jul 18 '15 at 8:45
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    $\begingroup$ @AntonFetisov Regarding Thm 10.5.6 - you said classical derived functors factor through Kan-extensions along the localization w.r.t to quasi-isomorphisms. Doesn't this only mean classical derived functors preserve q.i's? Are they not universal in this respect, or is the universality only there for the total derived functor? (p.s - I added the bit I forgot about taking the homology of the Kan extension.) $\endgroup$ – Arrow Jul 20 '15 at 8:40

EDIT Corrected a couple of inaccuracies and mistakes, added some references.

For the sake of clarity, let me work with non-negatively graded cochain complexes, and analyze the case of a left exact covariant functor, to prove that its right derived functor is a universal (covariant) cohomological $\delta$-functor.

Let $\mathcal A$ and $\mathcal B$ be two abelian categories, and let $F : \mathcal A \to \mathcal B$ be a left exact functor. Let me denote by $\gamma_\mathcal{A}$ the localization functor $\mathrm{Ch}_+(\mathcal{A}) \to \mathrm{D}_+(\mathcal{A})$. Let $\tilde S_\mathcal{A}$ denote the full subcategory of $\mathrm{Ch}_+(\mathcal A)$ consisting of discrete complexes (i.e. of complexes whose differentials are all zeroes). Let $S_\mathcal{A}$ denote its essential image in the derived category.

First of all, observe that a $\delta$-functor is a particular case of what MacLane calls "connected sequences" (connected sequence is a $\delta$-functor iff the long sequence it induces is exact). Now, defining a connected sequence is equivalent to define a functor $S_\mathcal{A} \to \mathcal{B}$ (see Proposition XII.8.1 and Proposition XII.8.2 in [1]). Part of the equivalence works as follows: given any $T : S_\mathcal{A} \to \mathcal{B}$, define $T^n(A) := T(A[n])$, and check that, given any exact sequence $0\to A\to B\to C\to 0$, the image of the class corresponding to it in $$S_\mathcal{A}(C[n],A[n+1]) \simeq \mathrm{D}_+(\mathcal A) (C[n], A[n+1]) \simeq \mathrm{Ext}^1_{\mathcal A}(C,A)$$ under $T$ gives you a morphism $\delta^n : T^n C \to T^{n+1} A$ giving $\{T^n\}$ the structure of a connected sequence. You can find the details of this construction in [1].

We have that $$\mathbb{R}F := \mathrm{Lan}_{\gamma_\mathcal{A}(-[0])} (\gamma_\mathcal{B} F(-)[0]).$$ Now, we can postcompose the bottom corners of the square defining $\mathbb{R}F$, along the functors $$\gamma_{\mathcal A} \circ \bigoplus_{n\geq 0} H^n : \mathrm{D}_+(\mathcal{A}) \to S_\mathcal{A}$$ and $$H^0 : \mathrm{D}_+(\mathcal{B}) \to \mathcal{B}$$ as follows

$$ \require{AMScd} \begin{CD} \mathcal{A} @>{F}>> \mathcal{B}\\ @VVV @VVV \\ \mathrm{D}_+(\mathcal{A}) @>{\mathbb{R}F}>> \mathrm{D}_+(\mathcal{B})\\ @VVV @VVV \\ S_\mathcal{A}@. \mathcal{B} \end{CD} $$

and further Kan extend, obtaining the connected sequence $RF : S_\mathcal{A} \to \mathcal{B}$ as $$\mathrm{Lan}_{\gamma_\mathcal{A} \circ \oplus_n H^n \circ (-)[0]}(H^0 \circ (-)[0] \circ F) \simeq \\ \simeq \mathrm{Lan}_{\gamma_\mathcal{A} (-[0])}(F).$$ Now, given any other connected sequence (in particular, any $\delta$-functor) $T : S_\mathcal{A} \to \mathcal{B}$, by definition of left Kan extension we have $$\mathrm{Nat}(RF,T) \simeq \\ \simeq \mathrm{Nat}(F, T \circ \gamma_{\mathrm{A}} (-[0])) \simeq \\ \simeq \mathrm{Nat}(F,T^0)$$ showing that the "universality" of $RF$ works not only for $\delta$-functors, but in general for connected sequences.

The fact that for $F$ left exact and $\mathcal A$ with enough injectives one has that $RF$ is not only a connected sequence, but really a $\delta$-functor, follows from XII.8.3, 4 and 5 in [1].

[1] MacLane - Homology

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