There are several models for Lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian fibrations, as described in Definition 4.1 of [Asaf Horev and Lior Yanovski: On Conjugates and Adjoint Descent][1]. An alternative construction is described in [David Gepner, Rune Haugseng, and Thomas Nikolaus: Lax Colimits and Free Fibrations in ∞-Categories](https://arxiv.org/pdf/1501.02161.pdf).

Similarly in the context of model categories, one can construct Lax limits using sections, as described in Definition 2.21 of  [Clark Barwick: On left and right model categories and left and right bousfield localizations](https://www.maths.ed.ac.uk/~cbarwick/papers/complete.pdf). For a similar account see Definition 3.1 in [Julia E. Bergner : Homotopy limits of model categories and more general homotopy theories][2]. 


Are all these models equivalent? In particular, is there an equivalence that allows me to move between Lax limit of a (small)diagram of model categories to the Lax limit of the corresponding diagram of presentable infinity categories. 

I suspect a similar assertion holds, based on remark 2.24 in the  paper above by Clark Barwick.  However, I have not been able to find a reference supporting this claim. Thanks in advance.

  [1]: https://arxiv.org/pdf/1705.04933.pdf
  [2]: https://drive.google.com/file/d/1gLAug9MKHmyQZuDuO7sk3WE_9oBDjlDK/view