8:18 AM
@AndreaMarino Don't you need at least that the inclusion z_0→Z is a cofibration?

1 hour later…
9:35 AM
Well, that's probably a good observation, it would be that Hom( , A) is a model category functor from Top^op to Top. Anyway I solved the case of Z=cube by a very ad-hoc method: the idea is to shrink the cube and make it "slide" through $z_0$, as the latter was the eye of a needle. I have in mind the image of $Z \times I^n \cup z_0 \times I^n \time I$ in mind to be extended to $Z \times I^{n+1}$
However, that was just for the sake of completeness, I solved the actual case :) if you come up with a cleaner "abstract" idea I would be happy to hear it anyway!