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Nov 13, 2019 at 20:13 comment added John Klein That is of course correct (both assertions).
Nov 13, 2019 at 15:15 comment added Julian Rosen The correspondence {universal covers} $\leftrightarrow$ {basepoints} depends on the choice of $g$, right? I agree $g$ is unique up to unique isomorphism, but it is not literally unique on the nose, and since there is only one isomorphism class in $\pi_X$, knowing $g(\tilde{X})$ only up to isomorphism is no information. I believe it is possible to choose $g$ to be constant on the level of objects.
Nov 13, 2019 at 14:22 history edited John Klein CC BY-SA 4.0
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Nov 13, 2019 at 13:02 history edited John Klein CC BY-SA 4.0
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Nov 13, 2019 at 12:54 history answered John Klein CC BY-SA 4.0