Coupling capacitor or differentiator.
Likely the latter, which, amplified by a logic gate, will act as a one-shot timer. Likewise C2 acts as a (much larger, longer time constant) coupling cap, presumably to affect DC offset (sync?) of the output.
Huh, and that's direct into the other gate's output? Will that do anything? ...Were 74LS outputs able to be dragged negative? They might be, I'm not sure offhand. Never tested it.
So, it's a bit of a weird circuit, but I guess if it works, it works. It surely won't work with 74HC, or probably other 74xx families; definitely use the LS as shown.
Note that, you still need a capacitor somewhere near the +5V/GND of this chip. It can be shared with others (in case this is a fragment of a larger schematic), or connected by a short length (say up to 10cm), but otherwise should be onboard, and maybe with an electrolytic (10uF say) too (say for connections over 50cm in length).
Tim