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 $\mathcal{C}/_T$.