Elemental Equilibrium

Here's my situation.

I'm an RF marooder and on my weapon that I use cyclone with I have lightning damage (2-23).
My map friend runs hatred and he mainly deals fire and cold damage.

Now here's my question. Will hatred make them resist cold, since it does more damage than lightning damage on cyclone, or will they still be vulnerable to cold and fire because the cold damage comes from an external source?

Heya there, I also have a RF char as a main and use EE.
You didnt mention if your friends runs EE??
I'm assuming not. Therefore If you are in range of his Hatred Aura then your cyclone will hit with Lightning/Cold Dmg, meaning the Mobs will have 50% REDUCED fire Res but 25% INCREASED Cold/Lightning Res. If you aren't in range of his aura, then you hit with just Lightning meaning the mobs will have lower Fire+Cold Res and Higher lightening res.
However if your friends runs EE then the last hit applied to the Mobs will take priority. The amount of damage has no significance.
Have a Read on EE here http://pathofexile.gamepedia.com/Elemental_Equilibrium

Best of Luck.

Hatred will mess up EE resist reduction for your friend. You'll be doing both cold and lightning damage, meaning when you hit an enemy they'll gain extra resists against those and lose resist against fire damage.

Depending on how much fire damage he deals though, it might not be that big of a deal. If, for example, he uses a 100% physical attack and his only elemental damage is from Hatred and Added Fire, he's probably losing nothing (or maybe gaining a bit of damage). If he's using Ice Crash however, you may have a problem.