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.

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.