So, I'm not that much of a Katarina user. I just got her because kitty kat Katarina was so darn fine.
Any tips on how I should go against Morgana?
Q and W a lot early if she tries to push. At lv 4+ just q/e/w her and you really can't lose, esp at 6. If you get caught by a binding then you'll take a lot of dmg, though.
You're better than her at pretty much all points of the game, if you're even or better, so you MAY want to play more passive if you are scared of their jungler.
Probably just farm. If you try and take her down by yourself she is going to black shield a lot of your damage and kill you with ult/binding/pool.
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