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.