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I’ve come across a category $\mathcal{C}$ recently with an object $T$ such that any other object $X$ has a map $f:X\rightarrow T$, and for any two maps $f,g:X\rightarrow T$, there exists a (not necessarily unique) automorphism $\sigma$ of $X$ such that $g=f\circ \sigma$. It’s easy to check that if a category has such $T$ with this property, it’s unique up to non unique isomorphism.

I’m wondering what the name for this property of $T$ (or $\mathcal{C}$) is, and what relation it implies between $\mathcal{C}$ and the slice category $\mathcal{C}/_T$.

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  • $\begingroup$ Does this question answer your question? $\endgroup$
    – varkor
    Commented Aug 28, 2023 at 23:08
  • $\begingroup$ Fantastic! Sorry for the dupe question. $\endgroup$
    – Chris H
    Commented Aug 28, 2023 at 23:15
  • $\begingroup$ @ChrisH it’s not a duplicate, notice that the linked question puts the automorphism in the other object. $\endgroup$
    – Fernando Muro
    Commented Aug 28, 2023 at 23:29
  • $\begingroup$ Oh true, thanks for pointing that out! $\endgroup$
    – Chris H
    Commented Aug 28, 2023 at 23:38
  • $\begingroup$ There is probably some description of things in terms of the forgetful fibration $\mathcal{C}/T\to\mathcal{C}$… $\endgroup$
    – Alec Rhea
    Commented Aug 28, 2023 at 23:39

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