I guess you want something along the lines of page 8 example of the notes perso.math.univ-toulouse.fr/btoen/files/2012/04/swisk.pdf

I don't claim I answered anything, so there is not much to debate. One could try proving a statement similar to arXiv:math/0603339 for relative derivators that are presentable biCartesian (to avoid the example of sections of a Cartesian, and not coCartesian, fibration over a simplex that can happen to be presentable). If that was done, then perhaps you could start with an algebraic derivator, check its local presentability so that it comes from a model-categorical bifibration and then use the above to not worry about anything. It seems like a lot of work, maybe something else was meant by DCC.

Doesn't seem likely in general? The nLab statement you are referring to is about Quillen bifunctor giving a Quillen bifunctor, but it is not happening in the way you are describing. As an example, take $SSet$ and $Map: SSet^{op} \times SSet \to SSet$. If we consider bisimplicial sets (=$\Delta^{op}$-diagrams in $SSet$), then the end construction applied to $Map$ gives the mapping space functor $\underline{Map}$ on bisimplicial sets. But usually to calculate $\mathbb R \underline{Map}(X,Y)$, one has to take a fibrant replacement of $Y$ which is not simply the pointwise fibrant replacement.

@TimCampion I dont though! :) The idea is to show that the functor $Fib_\lambda /f \subset (A^{[1]})_\lambda/f$ is cofinal, and this amounts to saying that a map between $\lambda$-presentable objects factors in the wfs with the middle object being $\lambda$-presentable. So some assumption is still needed on the wfs in question. Thanks for your idea it was of use indeed.

@TimCampion alright I think I see it; you assumed however that the factorisation corresponding to the wfs is accessible, I suppose?

@TimCampion could you please provide a reference for the statement that you mention? Namely that an accessible and accessibly embedded subcategory of $A^{[1]}$ is generated by a set of morphisms of $A$. I only find one for small injectivity-classes, which does not quite give the same. Thank you.

Tim: thank you, I think your comment solves one particular case I am looking in.

Thanks for the reference. They seem to assume that both $C \to D$ and $B \to D$ preserve colimits, which is not exactly my case. Do you reckon that the proof goes through still?

Ideally, no. The example I have in mind is the category of sections of a Grothendieck fibration $E \to [1]$ such that $E(0),E(1)$ are presentable and $E(1) \to E(0)$ is accessible, with no evident adjoint.