We know by Karol Szumiło's thesis (<https://arxiv.org/pdf/1411.0303.pdf>) that there is an equivalence between the two fibration categories of cofibration categories on one side and cocomplete $\infty$-categories on the other. By a dual argument, we obtain an equivalence between fibration categories and complete $\infty$-categories (I'm guessing, in the form of cofibration categories).


1. Does either of these constructions restrict to the case of combinatorial model categories vs. presentable $\infty$-categories? In other words, is it an (almost) immediate consequence of the main theorem therein that there is also an equivalence between some model-like structures of combinatorial model categories and presentable $\infty$-categories?
2. Similar question, but considering all model categories (or maybe just categories with both a fibration and a cofibration structure) on one side and complete/cocomplete $\infty$-categories on the other?



Correspondence between classes of model categories and classes of $\infty$-categories
As pointed out in the answer to [Localizing $\mathrm{CombModCat}$ at the Quillen equivalences](https://mathoverflow.net/questions/304399/localizing-mathrmcombmodcat-at-the-quillen-equivalences/405993#405993),
the answer to Question 1 whether



> 
> there is also an equivalence between some model-like structures of combinatorial model categories and presentable ∞-categories?
> 
> 
> 


is affirmative and is provided by the paper
[Combinatorial model categories are equivalent to presentable quasicategories](https://arxiv.org/abs/2110.04679).


The model-like structure used in the paper is that of the relative category.


Concerning Question 2:



> 
> Similar question, but considering all model categories (or maybe just categories with both a fibration and a cofibration structure) on one side and complete/cocomplete ∞-categories on the other?
> 
> 
> 


there are pretty severe size issues with defining “all model categories” or “all complete/cocomplete ∞-categories”.


If, however, we use universes to resolve these size issues,
then the answer is positive and can be deduced from Theorem 8.2 in the cited paper, in a manner similar to Proposition 8.3.


