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    $\begingroup$ In principle, there could be a problem. But I think usually there is no problem if you are careful enough to keep track of the universe parameters. $\endgroup$ Commented 2 days ago
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    $\begingroup$ They are certainly universe-dependent prima facie. The whole point is to show that they are not universe dependent despite initial appearances. $\endgroup$ Commented 2 days ago
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    $\begingroup$ @ZhenLin: My concern is that if in the middle of establishing some theorem about category theory, one passes to a larger universe, then usually one tacitly assumes that things which "were" true "stay" true. This is perhaps a naive viewpoint, but I could certainly imagine someone with less experience having it. I suppose what I'm really looking for is a deeper explanation/intuition about which statements "should" be universe independent, and which are not. $\endgroup$ Commented 2 days ago
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    $\begingroup$ @SimonHenry: In the examples of changing universe I am familiar with, this is in some sense just a technical device employed to make our categories small. But my question is whether passing to a larger universe could cause some other "change" which is undesirable or unwanted. I give an example above in my comment to Zhen Lin. $\endgroup$ Commented 2 days ago
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    $\begingroup$ This paper is all about the situation described in the beginning of the question: [2009.07164] Categorical large cardinals and the tension between categoricity and set-theoretic reflection arxiv.org/abs/2009.07164. $\endgroup$ Commented yesterday