# Homology of mapping telescope

It is stated here http://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf that if $X$ is an increasing union of the type $X=\bigcup_{i \in I}X_i$ (where $X_i \subset X_{i+1}$), then we have an isomorphism for any homology theory $E$:$$colim_i E_* (X_i) \simeq E_*(X).$$ In order to prove this, the mapping telescope of the inclusions $j_i: X_i \to X_{i+1}$ is introduced, throughout pages 115-116 we reach a diagram which should attest the existence of the desired iso. (The bottom row is the well-known presentation of the colimit of a functor towards abelian groups).

I cannot see why there should exist such an isomorphism $\xi$, and this is also a bit strange, since the author previously claimed that he wouldn't prove the claim directly. Does that map come from pure diagram chasing orare there involved the previous construction in a non-obvious way?

Any help will be highly appreciated, thanks in advance.
