Let $\mathit{Pr}^L$ be the $\infty$-category of presentable $\infty$-categories and continuous functors in some universe. Is it presentable itself a larger universe?



Is the $\infty$-category of presentable $\infty$-categories presentable?
When you pass to a larger universe, all categories that were in your old universe become small. A small category which is not a poset cannot be closed under colimits (think about taking coproducts with index sets larger than your category), and so cannot be presentable. All this applies equally well to $(\infty,1)$-categories, and ought to apply to any other reasonable generalized kind of category.


