This question is a crosspost of the second part of [this MSE question](https://math.stackexchange.com/questions/1284179/derived-functors-homotopical-vs-homological-approach).


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.




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**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?



Derived functors - homotopical vs homological approach
**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].




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[1] MacLane - Homology


