I'm trying to prove that a reduced homology theory can be define from an unreduced one, but the problem is to define a border map for the reduced homology using the unreduced homology groups.
You might know, Hatcher uses the maps
$\overline H_n(X)=\ker(H_n(X)\rightarrow H_n(point))$
$H_n(X)=\overline H_n(X\sqcup point)$
It's left as an exercise to check that these two maps are inverses of each other as transformation of the unreduced and the reduced homology theories.