I originally [posted this on Maths SE](https://math.stackexchange.com/questions/3322212/motivation-intuition-behind-the-definition-of-delta-functors-and-related-concept), but then realised that the question probably fits MO better, as my objective was to gain different perspectives regarding the matter.


1. Why are $\delta$-functors between abelian categories $\mathfrak{A}$ and $\mathfrak{B}$ defined as a collection of (right) derived functors, and connecting morphisms $\delta^i: T^i(A'') \rightarrow T^{i+1}(A')$ (given an original short exact sequence $0 \rightarrow A' \rightarrow A \rightarrow A'' \rightarrow 0$ in $\mathfrak{A}$)
$$(T^\*, \delta^\*) := \{(T^i, \delta^i)\}\_{i \geq 0}$$
Particularly, in the context of algebraic geometry, what about this definition is geometric ?
2. What is a geometric interpretation of the universality of $\delta$-functors ?
3. What is a geometric interpretation of effaceable $\delta$-functors, and by extension, theorem 2.2.2 in Grothendieck's Tohoku paper stating that effaceability implies universality ?



Motivation/intuition behind the definition of delta-functors and related concepts
I am not an expert on the subject but looking at the definition I would dare to say that these are basically the axioms that characterize an homology theory, if it weren't for the fact the $\delta$ maps usually go in the opposite directions.


The family of functors play the role of the various homology functors (one for each degree) and the $\delta$-maps are required to extend the family of mappings
$$T^n(A') \to T^n(A) \to T^n(A'')$$ 
(obtained applying the functors $T^n$'s to the two non trivial morphism of the short exact sequence)
to a long exact sequence.


The additional axioms on the data for a $\delta$-functor are required so that the long exact construction can be extended to a functor from the category of short exact sequences in the domain category $\mathcal{A}$ to a category of long exact sequences in the category $\mathcal B$.


If you take a look these are some of the Eilenberg-Steenrod's axioms for an homology theory: you are basically dropping the axioms for the excision which could be meaningless in a generic abelian category.


The universal property capture another property of homology theory: the fact that it is completely determinated by its 0-degree functor, indeed two homology theories satisfying the Eilenberg-Steenrod axioms that have naturally isomorphic 0-degree functors are isomorphic as homology theories.


Basically a universal $\delta$-functor seems to be a sort of homology that is completely determinated by its 0-degree part.


This should address part 1 and 2. Unfortunately I am not familiar with the Tohoku, so I can't address part 3, but hopefully someone better qualified than myself will be able to answer that.


Meanwhile I wish this was of some help.


