ated as ${\mathsf{Stab}_L}(\Phi)$ for very small classes $\Phi$ of functors such as $\{\emptyset\}$ and $\{\emptyset,\ulcorner\}$. Can they also be generated as ${\mathsf{Stab}_L}({\mathsf{Abs}_L}(\Upsilon))$ for “manageable” collections $\Upsilon$ of derivators? For instance, are there “universal” pointed or stable derivators that suffice to detect whether a given functor is absolute for all pointed or stable derivators?

To attack these questions, we use the technology of enriched derivators and weighted limits. We will see that it suffices to answer the first two questions positively, but it is not quite adequate for the third in general, although in particular cases the answer is yes. In [@gs:enriched] we will use a better technology to answer the third question positively in general as well.

Enriched derivators {#sec:enriched-derivators}
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We begin by defining the basic notions of enriched derivators. We freely make use of the language and techniques established in [@gps:additivity], in particular the language of *monoidal derivators* as it is developed in detail in [@gps:additivity §3]. In that paper there is also a detailed discussion of *two-variable adjunctions of derivators* [@gps:additivity §§8-9].

A monoidal derivator is a pseudo-monoid object in $\cDER$ (the 2-category of derivators and pseudonatural transformations) such that the monoidal structure $\otimes\colon{\sV}\times{\sV}\to{\sV}$ preserves colimits separately in both variables. The pseudo-monoid structure precisely amounts to a lift of ${\sV}\colon\cCat\op\to\cCAT$ against the forgetful functor from the $2$-category of monoidal categories, strong monoidal functors, and monoidal transformations. The resulting monoidal structures are denoted by $({\sV}(A),\otimes_A,\lS_A)$.

We will also have occasion to consider the following weaker notions.

A **left derivator** is a prederivator satisfying all the axioms of a derivator except the existence of right Kan extensions.

A morphism of left derivators is again a ps