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


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*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.

*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?

*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.
 A: For (2) I suggested a possible solution for this here: Lemma 5.4.5.11 of HTT.
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.
