Monstrosity of the Lake
When Monstrosity of the Lake enters, you may pay {5}. If you do, tap all creatures your opponents control, then put a stun counter on each of those creatures. (If a permanent with a stun counter would become untapped, remove one from it instead.) Islandcycling {2} ({2}, Discard this card: Search your library for an Island card, reveal it, put it into your hand, then shuffle.)
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Official rulings (2) · Scryfall
Typecycling is a form of cycling. Any ability that triggers on a card being cycled also triggers on a card being typecycled. Any ability that stops a cycling ability from being activated also stops a typecycling ability from being activated.
Unlike the normal cycling ability, typecycling doesn't allow you to draw a card. Rather, it lets you search your library for a card with the type or types indicated by the ability name. For example, a card with basic landcycling lets you search for a basic land card, and a card with Wizardcycling lets you search for a Wizard card.