Originally Posted by junja
so if theyre on the same team, morgana and kayle would ignore their differences?
but if theyre on opposite teams, garen and katarina will still hold a grudge?
I imagine it's more along these lines:
For Morgana and Kayle, the outcome of their battle could have dramatic effects on the politics and peace of the world they're in. Kayle, for sure, holds to high ideals, and would prioritize the battle at hand over her feud--partly because her grudge with Morgana is not nearly as all-consuming.
Morgana seethes with hatred for Kayle, but she could probably ultimately come to the same conclusion--even if she didn't care about the world she was fighting for, helping Kayle in this short battle is preferable to getting killed over and over (just because you revive doesn't mean dying doesn't hurt.) Furthermore, if the summoners suspected she was throwing the fight on account of her grudge, Morgana would have to face their wrath. The lore suggests that they are all very, very powerful mages in their own right; if nothing else, they could probably send Morgana back to where she came--or to somewhere worse.
For Garen and Katarina... the opportunity to beat up on your nemesis is probably a welcome perk to the job.
Also, you gotta remember: if someone is going to let personal grudges interfere with their performance, they won't be keeping the job as a champion much longer. If Morgana and Kayle are still champions, it's probably because they've proven their ability to work objectively.