Why are compactly generated $\infty$-categories closed under limits in $\operatorname{Cat}_{\infty}$?

Question

In Proposition 5.5.7.6 of Lurie's Higher Topos Theory, Lurie states that the $\infty$-category $\operatorname{Pr}^R_{\omega}$ of compactly generated $\infty$-categories and filtered-colimit-preserving right adjoints is closed under limits in $\operatorname{Cat}_\infty$. I would like to understand some detail of the proof.

Lurie says that most of the proposition follows from Theorem 5.5.3.18, which states that the inclusion $\operatorname{Pr}^R \subseteq \operatorname{Cat}_\infty$ is closed under limits. However, as $\operatorname{Pr}^R_{\omega}$ does not seem to be a full subcategory of $\operatorname{Pr}^R$, I don't understand why the result of 5.5.3.18 gives us that the adjunctions $F_\alpha: \mathcal{C}_\alpha \rightleftarrows \mathcal{C}: G_{\alpha}$ appearing in the proof of 5.5.7.6 correspond to morphisms in $\operatorname{Pr}^R_{\omega}$, i.e. why $F_{\alpha}$ preserves compact objects/$G_\alpha$ filtered colimits. Trying to trace back through the results leading up to 5.5.3.18, I get the impression that the fact that $G_{\alpha}$ is accessible in the proof of 5.5.3.18 indirectly relies on Proposition 5.4.7.7, which says that any right adjoint is accessible. However, this proposition uses that we may choose the regular kardinal $\kappa$ as high as we wish before showing that $G_{\alpha}$ is $\kappa$-continuous, and in particular cannot be applied if we wish to show this for $\kappa = \omega$.

Question: Why do the maps $G_{\alpha}: \mathcal{C} \to \mathcal{C}_{\alpha}$ appearing in the proof of 5.5.7.6 preserve filtered colimits?

Answer 1

By lemma 5.4.5.5., the forgetful functor $Pr^R_\omega\to Cat_\infty$ preserves pullbacks : the projection functors in the pullback preserve filtered colimits.

It's easy to prove that the same thing holds for arbitrary products ( a colimit in $\prod_\alpha C_\alpha$ is computed pointwise, which means precisely that each projection preserves it), and so by 4.4.2.7, the forgetful functor preserves all limits.

The main point being the following :

Given a diagram $F: I\to Cat_\infty$ where each $F(i)$ has $K$-indexed colimits and each $i\to j$ induces a $K$-indexed colimit preserving functor $F(i)\to F(j)$, then $\lim_{i\in I}F(i)$ has $K$-indexed colimits and each projection preserves them.

You prove this the same way, using 5.4.5.5. and 4.4.2.7 to the category of $\infty$-categories that have $K$-indexed colimits, and $K$-indexed colimit-preserving functors.