In Ayoub's thesis, *Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I)*, Ayoub proved that given a stable homotopical $2$-functor (**Definition 1.4.1**) $\mathrm{H}:\mathrm{Sch}/S \longrightarrow \mathfrak{TR}$ ($\mathfrak{TR}$: $2$-category of small triangulated categories), then one has a formalism of four operations $(f^*,f_*,f_!,f^!)$ (**Scholie 1.4.2**). In the proof (which is very long), he has to make an assumption on the category $\mathrm{Sch}/S$ over a base scheme $S$, namely, he considers only quasi-projective $S$-schemes. The reason for doing so is that any morphism inside $\mathrm{Sch}/S$ can be "smoothified", namely, admits a factorization into a closed immersion followed by a smooth morphism.

The difficulty in the proof, as he noted in his ICM2014 talk *A guide to ( ́etale) motivic sheaves*, is that the category of smoothifications of a morphism is not filtered, in constrast to the category of Nagata compactifications. So the technique is different, in particular, one obtains the well-defined property of proper push-forward $f_!$ before knowing the validity of proper base change (in fact, in his thesis, he proved only the **projective** base change but one can strengthen this to proper base change by Chow lemma).

So my question is:

How can one remove the quasi-projective assumption? More precisely, how can we move from quasi-projective $S$-schemes to separated $S$-schemes of finite type? Ayoub in loc.cit. noted that the existence of $f^!$ is local on source, so we may reduce to the case of $f$ being quasi-projective, what does he mean by this "local on source"?) Cisinski and Déglise, Triangulated categories of mixed motives seemed to achieve this goal (

**Theorem 2.4.50**). I guess their work relies on Ayoub's thesis but I cannot see the way Ayoub's**Scholie 1.4.2**is used in Cisinski & Déglise' book.Besides, I am interested in proper base change theorem in a greater manner. In the second chapter of his thesis, he defined the so-call notion

*stable homotopical algebraic derivator*and deduced various base change formulas. I wonder whether one can replace "projective" by "proper" in any base change formula of algebraic derivators?