At page 4 of Strom's "Modern Classical Homotopy Theory" there is a universal formulation of the algebraic closure of a field. You can read it here from google books.
Exercise 1.2a is then to convince yourself that the stated universal property does in fact characterize the algebraic closure of a field.
I cannot see why the $\bar f$ in the diagram should be unique. For example if $F=\mathbb R$ and $A=E=\mathbb C$, wouldn't the identity and complex conjugation both make the diagram commute?
Thanks in advance
P.S.: I haven't touch math in years, but always wanting to learn algebraic topology I decided to work through Strom's book for the fun of it; so I apologize if I'm missing something trivial. I tried to fish google for "universal property algebraic closure" but nothing came up.