The Stacks project

Lemma 3.10.2. Let $\alpha $ be an ordinal as in Lemma 3.10.1 above. The category $G\textit{-Sets}_\alpha $ satisfies the following properties:

  1. The $G$-set ${}_ GG$ is an object of $G\textit{-Sets}_\alpha $.

  2. (Co)Products, fibre products, and pushouts exist in $G\textit{-Sets}_\alpha $ and are the same as their counterparts in $G\textit{-Sets}$.

  3. Given an object $U$ of $G\textit{-Sets}_\alpha $, any $G$-stable subset $O \subset U$ is isomorphic to an object of $G\textit{-Sets}_\alpha $.

Proof. Omitted. $\square$

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