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Each magecraft ability has a different effect, although they all have the same trigger condition, whenever you cast or copy an instant or sorcery spell.

For example, if you control Archmage Emeritus and cast an instant or sorcery spell, Archmage Emeritus's magecraft ability will trigger and you will draw a card.

If an effect creates a copy of an instant or sorcery spell, this will also cause the magecraft ability to trigger.

If an effect creates multiple copies of an instant or sorcery spell, magecraft abilities trigger once for each copy created by the effect.

Some effects instruct you to copy an instant or sorcery card in a zone other than the stack. These copies do not cause magecraft abilities to trigger. However, most effects that do this also allow you to cast the copy, and casting the copy will cause magecraft abilities to trigger.

For Professor Onyx's last loyalty ability, "this process" is the following: First, the next opponent in turn order may choose a card in hand without revealing it, then each other opponent in turn order does the same. Then all the chosen cards are revealed and discarded at the same time and each opponent who didn't discard a card (whether they chose not to or had an empty hand) loses 3 life. This all will happen seven times, for a maximum loss of 21 life per opponent.

For Professor Onyx's second loyalty ability, if an opponent has multiple creatures with the greatest power, that player choose which one to sacrifice. The next opponent in turn order chooses which creature they are sacrificing, then each other opponent in turn order does the same, knowing the choices made before them. Then all chosen creatures are sacrificed at the same time.

If Professor Onyx seems familiar, it's probably because you've seen this type of planeswalker before.