In Lemma 5.4.5.11 of HTT, the proof given relies on Lemma 5.4.5.10. However it seems that Lurie applies Lemma 5.4.5.10, which requires the given simplicial set to be contractable, to an arbitrary $\kappa$-small simplicial set.


This seeming incongruity was pointed out in this [question](https://mathoverflow.net/questions/320575/stability-of-accessible-infty-categories-under-some-operations) on mathoverflow 2 years ago. However an answer was never given, and therefore I thought I might re-ask this in a new question (Let me know if there is a better way to re-ask an unanswered question).


Is there either   

a) a way to salvage the proof given, or  

b) a new proof which avoids the issue, or   

c) is the proof actually correct (and we're all being daft)



Lemma 5.4.5.11 of HTT
I think there is a typo in Lemma 5.4.5.11: $K$ is supposed to be $\tau$-small and not $\kappa$-small. Note that if $\tau < \kappa$ and $K$ is $\kappa$-small but not $\tau$-small then the statement of the lemma is simply false: e.g., set $\mathcal{I}=\mathcal{J}=K=\mathbb{N}$ to be the poset of natural numbers (with arrows pointing from small numbers to larger numbers) and take both $f$ and $p$ to be the identity. Set $\tau=\omega$ and $\kappa$ an uncountable cardinal. Then $\mathcal{I}\_{p/}$ is empty, and so certainly not $\tau$-filtered.


Note that the second statement in 5.4.5.11 is independent of this problem, and its proof does not use any size bound on $K$ (there is a small typo though in the first line of the proof of (2), "where $K$ is now $\kappa$-small and weakly contractible" -> $K$ should be $K'$.


Note also that 5.4.5.11 is cited two times in HTT, once in 5.4.5.12 and once in 5.4.6.5, but in both cases $K=\Delta^0$ and is in particular $\tau$-small.


The issue will hence in principle be resolved if we prove 5.4.5.11 under this modified assumption:


*Proof of 5.4.5.11(1) when $K$ is $\tau$-small*


Let $\tilde{q}:K' \to \mathcal{I}\_{p/}$ be a $\tau$-small diagram classifying a compatible pair of maps $q: K' \to \mathcal{I}$ and $q':K \star K' \to \mathcal{J}$. Since $\mathcal{I}$ is $\tau$-filtered we can find an extension $\overline{q}:(K')^{\triangleright} \to \mathcal{I}$ of $q$. To facilitate notation later on let us write $L:= (K')^{\triangleright}$ and let $l \in L$ be the vertex corresponding to the cone point of $(K')^{\triangleright}$. Now $q'$ and $f\overline{q}$ combine to give a map $r:[K \star K'] \coprod\_{K'}L \to \mathcal{J}$. Since $\mathcal{J}$ is $\tau$-filtered and $K, L$ are $\tau$-small we can find an extension of $r$ to a map
$$ \overline{r}:\Big[[K \star K']\coprod\_{K'} L \Big]^{\triangleright} \to \mathcal{J} .$$
Let $x := \overline{q}(l)$ and $\alpha: f(x) \to y$ be the arrow corresponding to the restriction of $\overline{r}$ to $\Delta^1 = \{l\}^{\triangleright} \subseteq L^{\triangleright}$. Since $f$ is $\kappa$-cofinal there exists an arrow $\beta: x \to z$ in $\mathcal{I}$ and a map $\eta:\alpha \to f(\beta)$ in $\mathcal{J}\_{f(x)/}$. Since the inclusion $\{l\} \subseteq L$ is right anodyne we have that $L\coprod\_{\{l\}} [\{l\} \star \Delta^0] \subseteq L \star \Delta^0$ is inner anodyne and so may now extend the map $L\coprod\_{\{l\}} [\{l\} \star \Delta^0] \to \mathcal{I}$ determined by $\overline{q}$ and $\beta$ to a map $\overline{q}\_{\beta}:L \star \Delta^0 \to \mathcal{I}$. Next since $L \coprod\_{\{l\}} \{l\}^{\triangleright} \subseteq L^{\triangleright} \subseteq \Big[[K \star K']\coprod\_{K'} L \Big]^{\triangleright} $ is a sequence of an inner anodyne map followed by a right anodyne map we may extend the map
$$ \Big[[K \star K']\coprod\_{K'} L \Big]^{\triangleright} \coprod\_{L} [L \star \Delta^0] \coprod\_{\{l\} \star \Delta^0} [\{l\}^{\triangleright} \star \Delta^0] = $$
$$ \Big[[K \star K']\coprod\_{K'} L \Big]^{\triangleright} \coprod\_{\big[L \coprod\_{\{l\}} \{l\}^{\triangleright}\big]} \Big[\big[L \coprod\_{\{l\}} \{l\}^{\triangleright}\big] \star \Delta^0\Big]\to \mathcal{J} $$
determined by $\overline{r}$, $f\overline{q}\_{\beta}$ and $\eta$ to a map
$$ \overline{r}\_{\eta}:\Big[[K \star K']\coprod\_{K'} L \Big]^{\triangleright} \star \Delta^0 \to \mathcal{J} .$$
The restriction of $\overline{r}\_{\eta}$ to $K \star K'\star \Delta^0$ and the restriction of $\overline{q}\_{\beta}$ to $K'\star \Delta^0$ now determine an extension of $\tilde{q}:K' \to \mathcal{I}\_{p/}$ to a map $K'\star \Delta^0 \to \mathcal{I}\_{p/}$, as needed.


