sticky delta - bit of clarification

http://www.columbia.edu/~mh2078/FX_Quanto.pdf says "as the exchange rate moves, the volatility of an option with a given strike is also assumed to move in such a way that the volatility skew, as a function of delta, does not move". By "skew" I take it to mean curve shape. By "function" I take it to mean a curve with delta as the x-axis.

If you refresh your USD/JPY vol surface every hour, you will see the surface moves.  Let's keep things simple and just focus on the 1-year smile curve. This curve moves but its shape stays fairly constant. It stays constant if you plot (imp) vol against delta. I believe the shape doesn't stay constant if you plot vol against strike. We say FX vol is sticky delta not sticky strike.