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

- 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.

canonicalcolimit, indexed by $D_{/d}$, is $\tau$-filtered -- the 1-categorical version of this argument is Prop 1.22 in Adamek and Rosicky,Locally Presentable and Accessible Categories. $\endgroup$strictlybigger than $\kappa$ such that $\mathcal C$ is still $\tau$-accessible. Then $\mathcal D$ is equivalent to the full subcategory of $\mathcal C$ consisting of the $\tau$-accessible objects. $\mathcal C$ has $\kappa$-filtered colimits and the $\tau$-accessible objects are closed under $\tau$-small colimits. So $\mathcal D$ is indeed closed under $\tau$-small $\kappa$-filtered colimits. It's the fact that $\tau$ is strictly bigger than $\kappa$ which ensures that $\mathcal D'$ is not arbitrary. $\endgroup$