Can a sheltie and a corgi get along well with cats or will they try to herd them

If you train them not to herd the cats you will probably be ok, not to mention they will learn that it is hard to herd a cat :) If you have other animals that you can train them to herd, that should satisfy the instinct.

They will quickly learn that herding cats is unrewarding. If there r more herdable animals around, I bet they will act like there r no cats. Teaching them not to chase is simple. Just say no if they try it. Most dogs who come to a house with adult cats will pretty much ignore the cats when they're indoors. Outdoors can be a different story, but dogs & cats who live together do not usually get involved in real chasing. I have a part Border Collie, though, who is really good at telling me when a cat is on the kitchen counter. I call her the cat cop.

I have a corgi mix who gets on fine with our 2 cats.

Shelties are very trainable, and can be taught what is and what is not appropriate to herd. Mine only chase cats that are outdoors and only if they run. If the cat stops, they lose interest. They have lived with cats indoors with no issues.

they will herd the cats into pens, tie them up, and shave off their fur to be sold to sweater makers. they will give you some of the profit.

The cats will not let them herd them.