I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories.


1. In Proposition 5.4.4.3, Lemma 5.4.4.2 is used, but I don't see how $\mathcal{D}^{/F(x)}$ matches the hypotheses thereof, in that it is not at all obvious to me that it be $\tau$-filtered and, moreover, that it admit $\tau$-small $\kappa$-filtered colimits, both of which are required in Lemma 5.4.4.2.
2. Lemma 5.4.5.11 uses Lemma 5.4.5.10, but this in turn requires that the simplicial set in the middle be weakly contractible, which is not in the case at issue. Is there a way to obviate this, or an alternative proof altogether to Lemma 5.4.5.11?
3. In Lemma 5.4.5.13 (and dually later on, in Corollary 5.4.6.7), the square


$\require{AMScd}$
\begin{CD}
 \mathcal{C}^{p/} @>>> \text{Fun}(K \times \Delta^1, \mathcal{C})\\
 @V V V @VV V\\
 \ast @>p>> \text{Fun}(K \times \{0 \}, \mathcal{C})
\end{CD}


is declared to be a pullback, which puzzles me, in that while it expresses the condition that the diagram restricted to $K \times \{ 0 \}$ is $p$ it says nothing about the restriction to $K \times \{ 1 \}$, which should be constant.


Answers to some or all these problems would be really appreciated.



Stability of accessible $\infty$-categories under some operations
For (2) I suggested a possible solution for this here: [Lemma 5.4.5.11 of HTT](https://mathoverflow.net/questions/386073/lemma-5-4-5-11-of-htt/386139#386139).


For (3) it really appears to be a typo and can be fixed as in the comment of dhy. For (1), as explained by Tim in the comments, there is actually no mathematical issue in the sense that $\mathcal{D}^{/F(x)}$ is in fact $\tau$-filtered and admits $\tau$-small $\kappa$-filtered colimits. The first property is by the fact that the presheaf $\mathrm{Map}(-,F(x))$ on $\mathcal{D}$ becomes representable in $\mathcal{C} = \mathrm{Ind}(\mathcal{D})$, which is equivalent to its unstraightening being $\tau$-filtered. The existence of $\tau$-small $\kappa$-filtered colimits follows as Tom explains from the fact that $\mathcal{D}$ can be taken to be the full subcategory of $\tau$-compact objects in $\mathcal{C}$ (this is maybe the main missing explanation in the proof as Jacob wrote it). Since $\mathcal{C}$ admits $\kappa$-filtered colimits by the choice of $\kappa$ and $\mathcal{D}$ is closed under any $\tau$-small colimit that exists in $\mathcal{C}$ it follows that $\mathcal{D}$ admits $\tau$-small $\kappa$-filtered colimits.


