I'm trying to prove (and understand) the compression criterion which states that a function $f\colon(I^n,\partial I^n,J^n)\to (X,A,x_0)$ represent zero in the relative homotopy group $\pi_n(X,A,x_0)$ iff it is homotopic relative $\partial I^n$ to a map $g\colon(I^n,\partial I^n,J^n)\to (X,A,x_0)$ with image contained in $A$. Note: $J^n$ here is $\partial I^n - I^{n-1}$ the same as in the book algebraic topology by Hatcher.
For the first implication: Let $g$ be such a map. Then by taking the composition of $g$ with the deformation retraction to any point that is mapped to the $x_0$ we obtain a homotopy $I^n\times I \to X$ from $g$ to the constant map. If $f$ is homotopic to $g$ relative $\partial I^n$ then $[f]=[g]$ and therefore $[f]=0$. This is how every proof I can find of this goes but it is not clear to me where exactly we use that $g(I^n)\subset A$.
The reverse implication is essentially achieved letting $H\colon I^n\times I\to X$ be a homotopy between $f$ and the constant map and then altering the domain of $H$ such that what was first the boundary of the 0-level becomes the boundary of every level while the original boundary together with the 1-level is taken to be the new 1-level. What confuses me about this part is that this modification to the domain doesn't seem to be homeomorphism.
A proof can be found here but what I wrote is based on Hatcher. https://amathew.wordpress.com/2010/10/11/the-compression-criterion/
Any clarification would be greatly appreciated.