TITLE: Functorial cones
QUESTION [6 upvotes]: This is probably more a reference request than a real question. I was studying dg-categories 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 dg-enhancement. 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 dg-enhancement is stressed? Thank you.

REPLY [4 votes]: 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 dg-functor from the (homotopy coherent) dg-category of morphisms.
Or, if you like, I wrote both a master and a PhD thesis centered on dg-categories:
https://anisama.files.wordpress.com/2019/04/tesi_mag.pdf (master thesis)
https://anisama.files.wordpress.com/2019/04/tesi.pdf (phd thesis)