This is probably more a reference request than a real question. I was studying dgcategories in order to understand how one can derive a functorial cone construction when a triangulated category (which for what concerns me is often $D^b(X)$, for some smooth projective variety $X$) has a dgenhancement. I looked only for some papers, but all the ones I found claim the existence of functorial cones without proving the claim. Can anyone point me to somewhere where this is proved and the geometric use of dgenhancement is stressed? Thank you.

$\begingroup$ Well, in derived categories, the cone functor is just the derived functor of the cokernel functor (in general it is a particular homotopy pushout). The choice of an enhancement (DG, $\infty$categories, model category, ...) gives you a precise strategy to construct derived functors so, in particular, also the cone functor. $\endgroup$ – Simone Virili Mar 26 '19 at 11:46
The cone construction can be written down very explicitly, just following the definition of mapping cone of chain complexes. Good sources are in my opinion:
https://arxiv.org/pdf/math/0401009.pdf Definition 3.7
https://arxiv.org/pdf/math/0210114.pdf paragraph 2.9, where it is discussed how the cone is functorial as a dgfunctor from the (homotopy coherent) dgcategory of morphisms.
Or, if you like, I wrote both a master and a PhD thesis centered on dgcategories: https://anisama.files.wordpress.com/2019/04/tesi_mag.pdf (master thesis) https://anisama.files.wordpress.com/2019/04/tesi.pdf (phd thesis)

$\begingroup$ Thank you very much, this is exactly what I was looking for! $\endgroup$ – Federico Barbacovi Apr 8 '19 at 10:01
A proof in the context of model categories can be found in Proposition 6.3.5 of Hovey's book Model Categories. You could easily rewrite the proof to work in the context of dgcategories, where it is actually easier.
EDIT: Here is a source that does it for dgcategories, in Section 4.1

$\begingroup$ Thank you for your answer. Yes, I read Toen's notes, but I didn't understand why that was supposed to be e functorial cone construction. Could you expand about it a little please? $\endgroup$ – Federico Barbacovi Mar 27 '19 at 10:36