## Question:

Let $D:A\to (X\downarrow C)$ be a $\kappa$-good $S$-tree rooted at $X$ for a collection of morphisms $S$ in $C$, where $\kappa$ is a fixed uncountable regular cardinal. Then according to the proof of Lemma A.1.5.8 of *Higher Topos Theory* by Lurie, for any $\kappa$-small downward-closed $B\subseteq A$, the colimit of the restricted diagram, $\varinjlim D|_B$ is $\kappa$-compact in $(X\downarrow C)$.

Why is this true? (It is stated without proof.)

## Definitions:

For your convenience, here are the definitions:

Recall that an object $X$ in $C$ is called $\kappa$-compact if $h^X(\cdot):=\hom(X,\cdot)$ preserves all $\kappa$-filtered colimits (where $\kappa$-filtered means "$<\kappa$"-filtered, since the terminology is different depending on the source).

Recall that an $S$-tree rooted at $X$ for a collection of morphisms $S$ in $C$ consists of the following data:

- An object $X$ in C (the root)
- A partially ordered set $A$ whose order structure is well-founded (the index)
- A diagram $D:A\to (X\downarrow C)$ such that given any element $\alpha\in A$, the canonical map $$\varinjlim D|_{\{\beta:\beta<\alpha\}}\to D(\alpha)$$ is the pushout of some map $U_\alpha\to V_\alpha\in S$.

We say that an $S$-tree is $\kappa$-good if for all of the morphisms $U_\alpha\to V_\alpha$ above, $U_\alpha$ and $V_\alpha$ are $\kappa$-compact, and such that for any $\alpha\in A$, the subset $\{\beta: \beta < \alpha \}\subseteq A$ is $\kappa$-small.

**Edit**: It's easy to reduce the proof to showing that $D(\alpha)$ is $\kappa$-compact, since projective limits of diagrams $B\to Set$ are $|Arr(B)|$-accessible (and therefore $\kappa$-accessible since $B$ is $\kappa$-small), we perform the computation for $I$ a $\kappa$-filtered poset, and $F:I\to C$, assuming that $D(\alpha)$ is $\kappa$-compact for all $\alpha\in B$:

$$\begin{matrix}\ \varinjlim_I Hom_C(\varinjlim_B D, F)&\cong&\varinjlim_I\varprojlim_{B^{op}} Hom_C(D,F)\\ &\cong& \varprojlim_{B^{op}}\varinjlim_I Hom_C(D,F)\\ &\cong& \varprojlim_{B^{op}} Hom_C(D,\varinjlim_IF)\\ &\cong& Hom_C(\varinjlim_B D,\varinjlim_IF) \end{matrix}$$

**Edit 2**: I think the above reduction actually won't work, since it doesn't use the hypothesis that B is downward-closed.

`$B = \{\beta \leq \alpha\}$`

, but not helpful. Want transfinite induction on downward-closed subsets. Choose an increasing sequence $B(\gamma)$ as in proof of Lemma A.1.5.11. Base: $B(0) = \emptyset$ and $D_{\emptyset} = X$. Next: $D_{B(\gamma)} \to D_{B(\gamma+1)}$ is a pushout of a pushout of $U_{\alpha_{\gamma+1}} \to V_{\alpha_{\gamma+1}}$. Limit ordinal $\lambda$: $D_{B(\lambda)}$ is a $\lambda < \kappa$-small colim of earlier. (c.f., pf of Lemma A.1.5.6) $\endgroup$