If they don't, return that card under your control with a finality counter on it

Ajani, the Greathearted Angrath, Captain of Chaos Arlinn, Voice of the Pack Ashiok, Dream Render Chandra, Fire Artisan Davriel, Rogue Shadowmage Domri, Anarch of Bolas Dovin, Hand of Control Gideon Blackblade Gideon, the Oathsworn Huatli, the Sun's Heart Jace, Arcane Strategist Jace, Wielder of Mysteries Jaya, Venerated Firemage Jiang Yanggu, Wildcrafter Karn, the Great Creator Kasmina, Enigmatic Mentor Kaya, Bane of the Dead Kiora, Behemoth Beckoner Liliana, Dreadhorde General Nahiri, Storm of Stone Narset, Parter of Veils Nicol Bolas, Dragon-God Nissa, Who Shakes the World Ob Nixilis, the Hate-Twisted Ral, Storm Conduit Saheeli, Sublime Artificer Samut, Tyrant Smasher Sarkhan the Masterless Sorin, Vengeful Bloodlord Tamiyo, Collector of Tales Teferi, Time Raveler Teyo, the Shieldmage Tezzeret, Master of the Bridge The Wanderer Tibalt, Rakish Instigator Ugin, the Ineffable Vivien, Champion of the Wilds Vraska, Swarm's Eminence

The other use of counter refers to physical counters or markers that are placed on cards to modify their characteristics in some way
Casting Serras Emissary might put a huge target on you, but it is an easy way to coax out a removal spell so you can push through an even bigger threat