You canâ€™t exert a creature unless an effect allows you to do so. Similar effects that â€œtap and freezeâ€ a creature (such as that of Decision Paralysis) donâ€™t exert that creature.
Some cards have abilities that trigger whenever you exert any creature. These abilities trigger when you exert that creature or any other creature you control.
If you gain control of another playerâ€™s creature until end of turn and exert it, it will untap during that playerâ€™s untap step.
If an exerted creature is already untapped during your next untap step (most likely because it had vigilance or an effect untapped it), exertâ€™s effect preventing it from untapping expires without having done anything.
All cards in the Amonkhet set that let you exert a creature let you do so as you declare it as an attacking creature, as do some of the cards in the Hour of Devastation set. You canâ€™t do so later in combat, and creatures put onto the battlefield attacking canâ€™t be exerted. Any abilities that trigger on exerting an attacking creature will resolve before blockers are declared.