coyotes mostly eat mice --as long as they are not mixed with dogs.
YES, in the winter they will seek out cats and even smaller dogs---a group of coyotes will kills large dogs--even bear hunting large dogs are easily killed by a pack of coyotes---they also will attack children.
cats love food, so traps are an easy method, but dealing with cats and that and the fact you will catch many raccoons while trying to catch a cat----normal people do not want to deal with disposing of the critters.
a 7# cat can literally fight a large hunting hound dog and WIN.
house cats are sickly creatures ----while wild ones are powerful predators.