What happens if both Slaking with "Lazy" and Garbodor with "Garbotoxin" in play? Do they cancel each other out or what?

It depends on the order in which each Ability gets activated. Both Lazy & Garbotoxin require a condition in order to activate (either being the Active Pokemon or having a Tool attached), so when one of the abilities is active the other one is prevented. But if the ability gets disabled (like if Slaking is no longer the Active Pokemon or if Garbodor loses its attached Tool), then the other ability can activate even if it was previously prevented.

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