It is not clear to me the role of the domain and target in the definition of [prederivators](https://webusers.imj-prg.fr/%7Egeorges.maltsiniotis/ps/m.pdf).


For instance, the classical references put the domain as $\mathit{Dia}$, others as $\mathit{Cat}$ itself.
Sometimes $\mathit{Dia}$ comes with some [conditions](http://user.math.uzh.ch/ayoub/PDF-Files/THESE.PDF), and sometimes not.


The target can also vary from the large CAT, or the 2-category of triangulated categories….


Grothendieck uses
$$
D: \mathit{Dia}^\circ \rightarrow \mathit{Cat}.
$$
*On suppose tout au moins que $\mathit{Dia}$ contient les ensembles ordonnés finis, que $\mathit{Dia}$ est stable par limites finies, par sommes finies. Qu'avec toute catégorie $C$ et toute sous-catégorie strictement pleine ouverte (resp. fermée) elle contienne la sous-catégorie complémentaire.*


Wikipedia says that the idea of this domain is to serve as "[indices](https://en.wikipedia.org/wiki/Derivator#Prederivators)".



> 
> What prevents us from using not a 2-sub-category of $\mathit{Cat}$ but the complete category as domain and target?
> 
> 
> 


Sorry if this question is too simple for the site. But can someone clean this up explaining the use of $\mathit{Dia}$ here?



Indexing categories of derivators
The idea is to have more freedom of expression. No one wants to work with a fixed $Dia$ for ever. In particular, $Dia$ is made more general to allow us to change $Dia$ at will, according to our needs, including to make weird technical choices in some proof (for instance, if you want that (pre)derivators collectively form a reasonnable $2$-category without using Grothendieck universes, you could be happy to know that $Dia$ is small, or you might want to avoid (pre)derivators taking values in categories which are not locally small because you love the Yoneda Lemma and you do not know how to prove that your main example has this property for arbitrary shapes of diagrams yet). However, we can define $Cat$-indexed derivators and say that the restriction to some category of diagrams $Dia$ satisfy the axiom of derivators are satisfied, and this is fine most of the time. But there are properties which can proved using only certain shapes of diagrams and it is natural to restrict the theory to the fragment which only deals with them. A nice example is [the universal property of bounded derived categories](https://arxiv.org/abs/1512.02691), mimicking Keller's approach via [towers of triangulated categories](https://webusers.imj-prg.fr/%7Ebernhard.keller/publ/dcp.pdf). The proof only uses finite diagrams. It is formal to deduce an analogous universal property for the $Cat$-indexed derivator associated to an exact category $\mathcal{E}$ (associating to each $C$ the bounded derived categories of $\mathcal{E}$-valued functors defined on $C$). However, this is a typical example where I do not see how to deduce formally that the version for $Cat$-indexed derivators implies the version for derivators indexed by finite partially ordered sets (if you need that the assignment $X\mapsto\mathbb{D}(X\times C)$ is a (stable) derivator on finite partially ordered sets for any small category $C$, it is not easy to produce such a thing from a (stable) derivator derfined on finite partially ordered sets only). Therefore, in this case, the version for finite diagrams only seems to be a more powerful statement.


