Thanks for viewing that. It is just to convert the future value to a present value. That’s the mapping part right there.
The forward is to buy EUR 100 in future (EUR +100 FV) in exchange for (selling) $130.xx US in future. What is EUR 100 in US PV? = EUR 100 * exchange rate * PV factor = $125 USD PV. In this way, the most important of the three mapping is converting the future buy of EUR into present USD. This is mapping the instrument (partially) to a long position on the spot rate.
Similarly, selling the $130 in FV = -125 in PV.
It is all to “map” the instrument onto the 3 factors in PV terms. Now risk becomes a function of stressing the three mapped-to-factors.