Do fish fish?

Hopefully this doesn’t come across as a joke, I know it’s a bit silly but I genuinely want to know and my google skills are failing me. My bf used the old saying “teach a man to fish” but instead he said “teach a fish to fish” in which we started to debate: when a fish hunts down and eats another fish is that fish fishing? Or is that just hunting? Is there an actual term for when fish do it?