> **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 $g = f \circ \alpha$.

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.

> **Question.** What are "nearly initial" objects really called?