I don't care if you guys are shooting them or not, I have coyote hunted before and had a blast. But for those saying that removing them can't possibly create more coyotes:
While you can temporally reduce their numbers by trapping they come back to pre-trapping levels within 8 months:
Lethal control for reducing carnivore populations is a contentious issue throughout the world. While computer simulations have been developed modeling the effects of population reduction on coyote (Canis latrans) population parameters, testing these hypotheses with empirical data from the field...
digitalcommons.unl.edu
And although you can reduce breeding by hunting and trapping, you open up that area to increased immigration from other dispersing coyotes in surrounding areas as well as increase the amount prey on the landscape which can lead to larger litter sizes (some studies have shown they could double or triple):
From my reading, it sounds like whatever individuals you kill will be at least replaced within a year unless you are trapping/shooting them all year long every year. Meaning from the "shoot them while im deer hunting" standpoint we are all doing basically nothing except collecting some sweet furs to swaddle our infants