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 $G\textit{-Sets}_\alpha$.