I am working on a project which requires that I calculate homotopy limits of homotopy theories (i.e. $(\infty,1)$-categories). It may be relevant that the homotopy limits which interest me are in the shape of towers; that is, the indexing category looks like $\cdots\rightarrow\cdot\rightarrow\cdot$. Because I am interested in some category-theoretic constructions (e.g. homotopy adjunctions between $(\infty,1)$-categories, Kan extensions, and homotopy (co)limits within $(\infty,1)$-categories), quasicategories seem like a natural choice for modeling $(\infty,1)$-categories, since they have the most well-developed category theory. However, I would be open to using a different model (e.g. Bergner's model structure on simplicial categories or Rezk's complete Segal spaces) if it proved more convenient.

Anyway, my question is simply how to calculate these homotopy limits. I am aware of Emily Riehl's proof that there is a model structure on the category of marked simplicial sets which is Quillen equivalent to the Joyal model structure on simplicial sets. This is nice, because the former model category is a simplicial model category, which the latter is not (although it is Cartesian closed). If it comes down to it, I should be able to do everything I want to do in this setting, bringing to bear the techniques developed in Riehl's book *Categorical Homotopy Theory* for calculating homotopy limits in simplicial model categories. But I'd like to know if there's a more straightforward approach which does not leave the Joyal model structure on simplicial sets.

Sticking with Riehl's approach to homotopy limits via weighted limits, is there a particular weight I should be using for calculating homotopy limits in $\mathbf{qCat_\infty}$ as a simplicial category (but again, not a simplicial model category)? I don't know that $N(\mathcal{D}/-):\mathcal{D}\to\mathbf{sSet}_\mathrm{Joyal}$ is cofibrant in $\left(\mathbf{sSet}_\mathrm{Joyal}\right)_\mathrm{proj}^\mathcal{D}$. Does anyone know if it is, or if not, what the cofibrant replacement looks like? Would we take the free groupoid of $\mathcal{D}/-$ before taking the nerve (just a guess)?

One last question: the theory developed by Riehl and Verity in their series of papers makes the 2-categorical approach to the study of $(\infty,1)$-categories appealing (i.e. working in the homotopy 2-category of the $(\infty,2)$-category of $(\infty,1)$-categories). Does anyone know if homotopy limits in $\mathbf{qCat_\infty}$ agree with $\mathbf{Cat}$-enriched (conical) limits in $\mathbf{qCat_2}$? That would be a useful result, but I don't think I've seen anything to that effect in Riehl's book.

Last note: one of the limits which interests me is of a tower of isofibrations, but I don't know that the morphisms involved in the second limit are inner fibrations (although the diagram is certainly pointwise fibrant).