Does the forgetful functor from presentable $\infty$-categories to $\infty$-categories preserve filtered colimits?

Question

Marc's answer to my previous question gives a way to compute colimits in the category of presentable $\infty$-categories and continuous functors, using the (discontinuous) right adjoints to those functors. But in particular it is not true that the forgetful functor from presentable $\infty$-categories and continuous functors, to all $\infty$-categories and functors, preserves all colimits.

Does it preserve all filtered colimits? I am sorry if the answer is easy to find in Lurie's textbook. I wasn't able to do so right away.

Answer 1

It doesn't. For instance, the $\infty$-category of spectra is the colimit of the tower

$$ \mathcal{S}_* \stackrel\Sigma\to \mathcal{S}_* \stackrel\Sigma\to ... $$

in $Pr^L$, but its colimit in $Cat_\infty$ is the Spanier-Whitehead category whose objects are formal desuspensions $\Sigma^{-n}X$ of pointed spaces. The latter category is stable and (uncountably) accessible but lacks some infinite colimits, like the sum of $\Sigma^{-n}S^0$ for $n\geq 0$.