Question:
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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:
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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.



K-good trees and K-compactness of colimits over K-small downwards-closed subposets (500 point bounty if answered by Midnight EST))
Does this work? 


To prove $D(\alpha)$ is $\kappa$-compact for all $\alpha$ in $A$, assume otherwise, that there exists some counterexample. Then, by the fact $A$ is well-ordered, there is a minimal counterexample (i.e., there is a minimal element $\alpha$ in the set of $\gamma \in A$ such that $D(\gamma)$ is not $\kappa$-compact). This means $D\_\beta$ is $\kappa$-compact for all $\beta \lt \alpha$. Since $\{\beta: \beta \lt \alpha\}$ has cardinality less than $\kappa$, we have that 


$$colim\_{\beta: \beta \lt \alpha} D(\beta)$$ 


is $\kappa$-compact. Now, given a diagram of the form


$$V\_\alpha \leftarrow U\_\alpha \to colim\_{\{\beta: \beta \lt \alpha\}} D(\beta)$$


in the category of $\kappa$-compact objects, its pushout is also $\kappa$-compact. But the hypothesis is that $D(\alpha)$ is the pushout for some such diagram, so $D(\alpha)$ is $\kappa$-compact, and we have reached a contradiction. 


So $D(\alpha)$ is $\kappa$-small for all $\alpha \in A$. It follows that $colim\_{\beta \in B} D(\beta)$ is $\kappa$-compact for any subposet $B \subseteq A$ whenever this is a $\kappa$-small colimit. (The restriction to downward-closed $B$ is not much loss of generality, because if $B \subseteq A$ is full, then the colimit over such a $B$ is isomorphic to the colimit over its downward closure, since $B$ is cofinal in its downward closure.) 


