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> **Definition.** Call an object $X$ of a category $\mathbf{C}$ *nearly initial* iff firstly, it is weakly initial, and secondly, for all objects $Y$ and all morphisms $f,g : X \rightarrow Y$, there exists an automorphism $\alpha$ of $X$ such that $g = f \circ \alpha$.
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It's straightforward to show that any two nearly initial objects of a category $\mathbf{C}$ are isomorphic (though not necessarily up to unique isomorphism.) For instance, if I'm not mistaken, the algebraic closure of a field is, by definition, the unique (up to isomorphism) nearly-initial algebraically-closed extension of that field.



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> **Question.** What are "nearly initial" objects really called?
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What are "nearly initial" objects really called?
In Definition 4.1 of their paper [*Shapely monads and analytic functors*](https://arxiv.org/pdf/1512.05980.pdf), Richard Garner and Tom Hirschowitz call such an object a "Galois object".


