Lemma 3.10.1. With notations $G$, $G\textit{-Sets}_\alpha$, $\text{size}$, and $Bound$ as above. Let $S_0$ be a set of $G$-sets. There exists a limit ordinal $\alpha$ with the following properties:

1. We have $S_0 \cup \{ {}_ GG\} \subset \mathop{\mathrm{Ob}}\nolimits (G\textit{-Sets}_\alpha )$.

2. For any $S \in \mathop{\mathrm{Ob}}\nolimits (G\textit{-Sets}_\alpha )$ and any $G$-set $T$ with $\text{size}(T) \leq Bound(\text{size}(S))$, there exists an $S' \in \mathop{\mathrm{Ob}}\nolimits (G\textit{-Sets}_\alpha )$ that is isomorphic to $T$.

3. For any countable index category $\mathcal{I}$ and any functor $F : \mathcal{I} \to G\textit{-Sets}_\alpha$, the limit $\mathop{\mathrm{lim}}\nolimits _\mathcal {I} F$ and colimit $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} F$ exist in $G\textit{-Sets}_\alpha$ and are the same as in $G\textit{-Sets}$.

Proof. Omitted. Similar to but easier than the proof of Lemma 3.9.2 above. $\square$

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