Reduced homology and colimits

Question

I would like to prove that colimits commute with reduced homology, i.e that if $ X = \operatorname{colim}\limits_{n \in \Bbb{N}} X_n$, then

$$ \displaystyle\tilde H_k(X) = \operatorname{colim}\limits_{n \in \Bbb{N}} \tilde H_k(X_n). $$

Moreover, I would really like to deduce this only from the Eilenberg-Steenrod axioms. Do you think this is possible ?

Thanks.