/sci/ - Science & Math

I'm having trouble understanding the diagram chase in https://math.stackexchange.com/a/1999115
Specifically the part where the equality of [eqn](X \times_{X/R}A) \times_A (X \times_{X/R}A)'\rightrightarrows X\xrightarrow{f}Z[/eqn] cancels to equality of[eqn](X \times_{X/R}A) \times_A (X \times_{X/R}A)'\rightrightarrows X\xrightarrow{p}X/R[/eqn].
The answer explains this by saying that
>As [math]R[/math] is defined by [math]f[/math], we find then that the morphisms [math](X \times_{X/R}A) \times_A (X \times_{X/R}A)' \rightrightarrows X[/math] factor through [math]R[/math].
but I don't see how this can be justified without assuming what you wish to prove (that [math]g[/math] can be cancelled in the factoring [math]f=g\circ p[/math]).