At close range for coyotes almost anything will do but I'd list my minimum as a 22 mag. Long range my minimum would be a 223. The biggest caliber in my collection I'd reach for if I were going on a coyote hunt? 243. But in a target of opportunity situation I will shoot a coyote with whatever I'm carrying. A 223 is probably perfect for yotes however.
For deer my minimum is a 243. Maximum? Whatever, as long as you shoot it well. I personally would not go larger than a 7mm mag and consider 300 magnums and up overkill for our Alabama deer. But whatever blows your skirt up. It's not like a 375 H&H won't kill a deer. I just wouldn't want to use it for that purpose.
I have no experience with Elk or Grizzly and I'll leave that to people who do.
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