Why two simplices become equal in the colimit

Question

The following is an image of a proof found in Hovey's Model Categories, for which my question concerns just the second paragraph.

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I do not see the justification for the statement that we can find $\alpha(i,x)$ so that $gd_i{x}$ and $d_ig{x}$ become equal in $X_{\alpha(i,x)}$, i.e., that the two become equal in the colimit. Could someone please explain this?

Answer 1

The answer is essentially that this is because $i_\alpha g = f$, and $f$ and $i_\alpha$ are maps of simplicial sets.

Therefore, if we fix $i$ and $x$, $$i_\alpha (gd_ix)=fd_ix=d_ifx = d_i(i_\alpha g)x = i_\alpha(d_igx).$$ In other words, $gd_ix$ and $d_igx$ are equal in the colimit, so we can find some $\alpha(i,x)$ as claimed.