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?



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



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> 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.
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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.


