How does a "speed up capacitor" work in Schmitt trigger?
The positive feedback path provides the hysteresis that makes it a Schmitt trigger.  However, it also forms an RC filter between the parasitic capacitance C1 and the resistor R1||R2.  It's possible the hysteresis may not kick in quickly enough for some applications because of this delay.

C2 provides a momentary, relatively high current feedback path to make sure the positive input responds quickly enough to the output.