Given a locally presentable ($\infty$,1)category $C$, can the fibrewise stabilization of it's codomain fibration, also called its tangent category $TC$, be given in terms of the Grothendieck construction? Specifically, is there a natural choice of functor $F:C\to \infty\operatorname{Cat}$ such that $TC\simeq \int F$? Is it possible in the 1categorical case, where instead the tangent category of $C$ is the category of abelian groups internal to $C^{\Delta [1]}$, the arrow category of $C$? The codomain fibration can of course be written as the Grothendieck construction applied to the functor $C_{/()} : C \to n\operatorname{Cat}$ sending objects to corresponding slice categories, and I was hoping to paste some pullbacks to abelianize this, but I was having no luck.
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$\begingroup$ Well, every fibration over $C$ is the Grothendieck construction of some functor, since the Grothendieck construction is an equivalence. I thought the fibers of $TC$ were the stabilizations of the fibers of $C$? $\endgroup$ – Mike Shulman Jan 2 '17 at 22:18

$\begingroup$ So the corresponding functor into the appropriate category of fibres would simply send each object to the category of abelian group structures on it? This is the same functor which sends each object to the category of functors from each object's representable presheaf to the forgetful functor from Ab to Set, viewed as a hom in the overcategory nCat/Set? (Or, whatever the appropriate cosmos is for the category in question.) $\endgroup$ – Mathemologist Jan 3 '17 at 19:26

1$\begingroup$ No, it would send each object to the category of abelian group structures on objects of its slice category. $\endgroup$ – Mike Shulman Jan 3 '17 at 20:51

$\begingroup$ Oh, right. I forgot to pass to the arrow category before abelianizing. Thank you! $\endgroup$ – Mathemologist Jan 3 '17 at 21:04
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See the proof of Proposition 1.1.9 here https://arxiv.org/pdf/0709.3091v2.pdf.